On Spanning Trees with few Branch Vertices
On Spanning Trees with few Branch Vertices Warren Shull Emory University Joint work with Ron Gould
May 21, 2017
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Spanning trees
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Spanning trees Leaf of a tree: degree 1
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 )
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 )
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways:
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
In the next few slides, spanning trees are more “desirable” the fewer branch vertices they have.
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
In the next few slides, spanning trees are more “desirable” the fewer branch vertices they have. What conditions might lead to a desirable spanning tree? Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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One possible condition: independent sets
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One possible condition: independent sets A desirable spanning tree is reached by adding edges
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more...
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not?
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges. And of course...it helps if the graph is claw-free.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges. And of course...it helps if the graph is claw-free. What are the best possible parameters? Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012)
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices...
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
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k +1 Km
Km
Km
Km
Km
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Km
Km
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others... ...for a minimum of k + 1 branch vertices. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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k +1 Km
Km
Km
Km
Km
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Km
Km
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| =
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| =
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3)
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3)
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
independent set
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
independent set X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X |
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3 + k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3 + k = 2k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | = k + 3 + k = 2k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
3 + 3k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
= mk + 3m + 2k
3 + 3k 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
= mk + 3m + 2k
3 + 3k 3 = |V (G )|
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3
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Corollary Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves, or an independent set of 3 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Corollary Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves (0 branch vertices), or an independent set of 3 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Matsuda, Ozeki, Yamashita 2012) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 1 branch vertex, or an independent set of 5 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges.
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof:
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph.
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. By the theorem of Kano et. al. above (with k = 4), G has a spanning tree with at most 6 leaves. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Among spanning trees with at most 6 leaves, choose a tree T such that:
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible.
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1).
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible.
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible. How many di↵erent structures could T possibly have?
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First case: T has only 5 leaves (the fewest possible):
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First case: T has only 5 leaves (the fewest possible):
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First case: T has only 5 leaves (the fewest possible):
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Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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(T3) If choosing between trees of these two types, we always choose one of the first type. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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path path
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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First case:
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First case:
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X Partition the tree
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First case: Choose independent set X Partition the tree
M1 M4
M2
M3 Q1
Q2
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5
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First case:
M1 M4
M2
M3 Q1
Q2
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M5
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First case: Consider one part
M1 M4
M2
M3 Q1
Q2
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5
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First case: Consider one part
M1
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First case: Consider one part Show certain neighbor sets must be disjoint
M1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
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b2
l5
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 )
l5
j 2 {1, 2}
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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Disjoint sets:
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
v 2X
|NG (v ) \ V (M1 )|
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )| |NG (li ) \ V (M1 )| +
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
2 X j=1
|NG (bj ) \ V (M1 )|
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )| |NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| +
X i6=1
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| +
2 X j=1
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )|
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )|
=
(X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
|NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| + (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1
X i6=1
+
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| + X i6=1
2 X j=1
|NG (li ) \ V (M1 )| +
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )| 2 X j=1
|NG (bj ) \ V (M1 )|
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )|
=
(X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
|NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| + (NG (l1 ) \ V (M1 ))
|V (M1 )|
j 2 {1, 2} i 6= 1
X i6=1
+
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| + X i6=1
2 X j=1
|NG (li ) \ V (M1 )| +
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )| 2 X j=1
|NG (bj ) \ V (M1 )|
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l1
l4
l3
b1
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l2
b2
l5
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l1
l3
l2
M1
l4
b1
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b2
l5
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l1
l3 M1
M2
M3
M4 l4
l2
Q1
Q2
b1
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M5 b2
l5
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l1
l3 M1
M2
M3
M4
Q1
l4
l2
Q2
b1 X
v 2X
M5 b2
l5
|NG (v ) \ V (M1 )| |V (M1 )|
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l1
l3 M1
M2
M3
M4
Q1
l4
l2
Q2
b1 X
v 2X
|NG (v ) \ V (Mh )| |V (Mh )|
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M5 b2
l5
h 6= 3
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2 z
|NG (v ) \ V (Mh )| |V (Mh )|
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M5 b2
l5
h 6= 3
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2 z
|NG (v ) \ V (Mh )| |V (Mh )|
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M5 b2
l5
h 6= 3
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l1
l3 M1
z
Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2
z
|NG (v ) \ V (Mh )| |V (Mh )|
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M5 b2
l5
h 6= 3
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l1
l3 M1
z
Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
Q2
z
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
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l1
l3 M1
z
Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
Q2
z
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
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l1
l3 M1 Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }|
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j 2 {1, 2}
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }|
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j 2 {1, 2}
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l1
l3 M1 Q1
v 2X
z
Q2
M5
w1 z w2
b1 X
M2
M3
M4 l4
l2
b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }| X
v 2X
|NG (v )| |V (G )|
j 2 {1, 2}
3
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
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FUTURE WORK
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing).
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time.
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done:
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial.
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens).
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices. Can it be done for certain classes of graphs?
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices. Can it be done for certain classes of graphs? And/or within some margin?
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Thank you for your attention!
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May 21, 2017
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Spanning trees
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Spanning trees Leaf of a tree: degree 1
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 )
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 )
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways:
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
In the next few slides, spanning trees are more “desirable” the fewer branch vertices they have.
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Spanning trees Leaf of a tree: degree 1 Branch vertex of a tree: degree
3
Hamiltonian paths are a special kind of spanning tree Max degree 2 (except K2 and K1 ) 2 leaves (except K1 ) No branch vertices
Some spanning trees are “close” to being a Hamiltonian path, in a few di↵erent ways: Low maximum degree Few leaves Few branch vertices
In the next few slides, spanning trees are more “desirable” the fewer branch vertices they have. What conditions might lead to a desirable spanning tree? Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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One possible condition: independent sets
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One possible condition: independent sets A desirable spanning tree is reached by adding edges
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more...
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not?
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges. And of course...it helps if the graph is claw-free.
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One possible condition: independent sets A desirable spanning tree is reached by adding edges A large independent set is reached by removing edges Given the right parameters, one or the other must exist. But there’s more... Independent sets may have many outgoing edges. Can we choose one that does not? We can if we remove enough edges!
Given the right parameters, there is either a desirable spanning tree or a large independent set with few outgoing edges. And of course...it helps if the graph is claw-free. What are the best possible parameters? Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012)
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices,
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices...
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
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k +1 Km
Km
Km
Km
Km
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Km
Km
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others...
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k +1 Km
Km
Km
Km
Km
Km
Km
Connected and claw-free Any spanning tree must have a branch vertex in this triangle... ...and each of these others... ...for a minimum of k + 1 branch vertices. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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k +1 Km
Km
Km
Km
Km
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Km
Km
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| =
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| =
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3)
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3)
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
independent set
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k
independent set X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X |
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3 + k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | k + 3 + k = 2k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k |X | = k + 3 + k = 2k + 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1)
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
3 + 3k
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
= mk + 3m + 2k
3 + 3k 3
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k +1 Km
Km
Km
Km
Km
Km
Km
|V (G )| = m(k + 3) + 2k = mk + 3m + 2k X
|X | = k + 3 + k = 2k + 3
deg(x)
(k + 3)(m
1) + 3k
x2X
= mk
k + 3m
= mk + 3m + 2k
3 + 3k 3 = |V (G )|
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3
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
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Theorem (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices. This is best possible.
Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Corollary Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves, or an independent set of 3 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Corollary Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 leaves (0 branch vertices), or an independent set of 3 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Matsuda, Ozeki, Yamashita 2012) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 1 branch vertex, or an independent set of 5 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
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Conjecture (Matsuda, Ozeki, Yamashita 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k branch vertices, or an independent set of 2k + 3 vertices with at most |V (G )| 3 outgoing edges. This is best possible.
Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges.
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof:
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph.
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Theorem (Kano, et. al. 2012) Let k be a non-negative integer and let G be a connected claw-free graph. Then G contains either a spanning tree with at most k + 2 leaves, or an independent set of k + 3 vertices whose degrees add up to at most |V (G )| k 3.
Theorem (Gould, S. 2017) Let G be a connected claw-free graph. Then G contains either a spanning tree with at most 2 branch vertices, or an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. Proof: Let G be a connected claw-free graph. By contradiciton, assume G has neither a spanning tree with at most 2 branch vertices, nor an independent set of 7 vertices with at most |V (G )| 3 outgoing edges. By the theorem of Kano et. al. above (with k = 4), G has a spanning tree with at most 6 leaves. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Among spanning trees with at most 6 leaves, choose a tree T such that:
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible.
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1).
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible.
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Among spanning trees with at most 6 leaves, choose a tree T such that: (T1) T has as few branch vertices as possible. (T2) T has as few leaves as possible, subject to (T1). (T3) TBA (T4) The parts of T in-between branch vertices are as small as possible. How many di↵erent structures could T possibly have?
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First case: T has only 5 leaves (the fewest possible):
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First case: T has only 5 leaves (the fewest possible):
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First case: T has only 5 leaves (the fewest possible):
path
path
path
path
path
path
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
path
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
path
path
path
path
path
path
path
path
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
path
path
path
path
path
path
path
path
path
path
path
path
path
path
path
path
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
path
path
path
path
path
path
path
path
path
path
path
path
path
path
path
path
(T3) If choosing between trees of these two types, we always choose one of the first type. Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
path path
path path
path path
path
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path path
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
path path
path path
path
path path
path
path path
path
path path
path
path
path
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
path
path
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First case:
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First case:
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X
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First case: Choose independent set X Partition the tree
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First case: Choose independent set X Partition the tree
M1 M4
M2
M3 Q1
Q2
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5
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First case:
M1 M4
M2
M3 Q1
Q2
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M5
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First case: Consider one part
M1 M4
M2
M3 Q1
Q2
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5
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First case: Consider one part
M1
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First case: Consider one part Show certain neighbor sets must be disjoint
M1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
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b2
l5
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 )
l5
j 2 {1, 2}
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 )
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l3
l2
M1
l4
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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First case: Consider one part Show certain neighbor sets must be disjoint
l1
l4
l3
l2
b1
b2 NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
l5
j 2 {1, 2} i 6= 1
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Disjoint sets:
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1
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May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
v 2X
|NG (v ) \ V (M1 )|
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )| |NG (li ) \ V (M1 )| +
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
2 X j=1
|NG (bj ) \ V (M1 )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )| |NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| +
X i6=1
j 2 {1, 2} i 6= 1 (X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| +
2 X j=1
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )|
May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )|
=
(X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
|NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| + (NG (l1 ) \ V (M1 ))
j 2 {1, 2} i 6= 1
X i6=1
+
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| + X i6=1
2 X j=1
|NG (li ) \ V (M1 )| +
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )| 2 X j=1
|NG (bj ) \ V (M1 )|
May 21, 2017
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Disjoint sets: NG (bj ) \ V (M1 ) NG (li ) \ V (M1 ) (NG (l1 ) \ V (M1 )) X
=
v 2X 5 X i=1
|NG (v ) \ V (M1 )|
=
(X = {l1 , l2 , l3 , l4 , l5 , b1 , b2 })
|NG (li ) \ V (M1 )| +
= |NG (l1 ) \ V (M1 )| + (NG (l1 ) \ V (M1 ))
|V (M1 )|
j 2 {1, 2} i 6= 1
X i6=1
+
2 X j=1
|NG (bj ) \ V (M1 )|
|NG (li ) \ V (M1 )| + X i6=1
2 X j=1
|NG (li ) \ V (M1 )| +
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
|NG (bj ) \ V (M1 )| 2 X j=1
|NG (bj ) \ V (M1 )|
May 21, 2017
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l1
l4
l3
b1
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
l2
b2
l5
May 21, 2017
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l1
l3
l2
M1
l4
b1
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
b2
l5
May 21, 2017
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l1
l3 M1
M2
M3
M4 l4
l2
Q1
Q2
b1
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5 b2
l5
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l1
l3 M1
M2
M3
M4
Q1
l4
l2
Q2
b1 X
v 2X
M5 b2
l5
|NG (v ) \ V (M1 )| |V (M1 )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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l1
l3 M1
M2
M3
M4
Q1
l4
l2
Q2
b1 X
v 2X
|NG (v ) \ V (Mh )| |V (Mh )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5 b2
l5
h 6= 3
May 21, 2017
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2 z
|NG (v ) \ V (Mh )| |V (Mh )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5 b2
l5
h 6= 3
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2 z
|NG (v ) \ V (Mh )| |V (Mh )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5 b2
l5
h 6= 3
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l1
l3 M1
z
Q1 b1 X
v 2X
M2
M3
M4 l4
l2
Q2
z
|NG (v ) \ V (Mh )| |V (Mh )|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
M5 b2
l5
h 6= 3
May 21, 2017
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l1
l3 M1
z
Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
Q2
z
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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l1
l3 M1
z
Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
Q2
z
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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l1
l3 M1 Q1 b1 X
v 2X
X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
l5
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
j 2 {1, 2}
May 21, 2017
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l1
l3 M1 Q1 b1 X
v 2X
M2
M3
M4 l4
l2
z
Q2
w1 z w2
M5 b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }|
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
j 2 {1, 2}
May 21, 2017
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l1
l3 M1 Q1
v 2X
z
Q2
M5
w1 z w2
b1 X
M2
M3
M4 l4
l2
b2
|NG (v ) \ V (Mh )| |V (Mh )|
X
v 2X
h 6= 3
|NG (v ) \ V (M3 )| |V (M3 ) \ {z }|
X
v 2X
l5
|NG (v ) \ V (Qj )| |V (Qj ) \ {wj }| X
v 2X
|NG (v )| |V (G )|
j 2 {1, 2}
3
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Second and third cases: T has 6 leaves, but only 3 branch vertices.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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Fourth and fifth cases: T has 4 branch vertices (and therefore 6 leaves)
P
(T5) P is as short as possible, subject to (T1)-(T4).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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FUTURE WORK
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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FUTURE WORK We think we’ve proven the entire conjecture (currently editing).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
161 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
162 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done:
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
163 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
164 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens).
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
165 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices.
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
166 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices. Can it be done for certain classes of graphs?
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
167 / 169
FUTURE WORK We think we’ve proven the entire conjecture (currently editing). Algorithmically, we suspect we can guarantee either the spanning tree or the low-degree independent set in polynomial time. Open question once this is done: Reduce the degree of this polynomial (will likely be in the teens). This algorithm might not find the tree with the fewest branch vertices. Can it be done for certain classes of graphs? And/or within some margin?
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
168 / 169
Thank you for your attention!
Warren Shull Emory University Joint work On withSpanning Ron Gould Trees with few Branch Vertices
May 21, 2017
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