Partial Characterizations of Circle Graphs - Semantic Scholar
Partial Characterizations of Circle Graphs ´ Luciano N. Grippo and Mart´ın D. Safe Flavia Bonomo, Guillermo Duran, CONICET and Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires Departamento de Ingenier´ıa Industrial, Universidad de Chile Instituto de Ciencias, Universidad Nacional de General Sarmiento
Permutation graphs
Circle Graphs A circle graph is the intersection graph of a family of chords on a circle. More precisely, a graph G = (V , E) is a circle graph if it is the intersection graph of a family {Lv }v ∈V of chords on a circle; i.e., v, w ∈ V are adjacent if and only if Lv ∩ Lw 6= ∅ and v 6= w . The family {Lv }v∈V is called a circle model of G.
circle model circle graph
P4-tidy graphs
Bouchet’s Characterization I
I
A graph is a comparability graph if its edges can be transitively oriented. A graph G is a permutation graph if G and G are comparability graphs. A characterization of comparability graphs by Gallai (1967) leads immediately to a forbidden induced subgraph characterization of permutation graphs. We denote by G1 + G2 the join graph of G1 and G2, where V (G1 + G2) = V (G1) ∪ V (G2) and E(G1 + G2) = E(G1) ∪ E(G2) ∪ {uv : u ∈ V (G1) and v ∈ V (G2)}. Theorem (Golumbic, 1980). The join G = G1 + G2 is a circle graph if and only if both G1 and G2 are permutation graphs. The above result is crucial in proving the following two characterizations.
Cographs are the P4-free graphs, that is, graphs with no induced path of four vertices. It is well-known that cographs are circle graphs. I Let G be a graph and let A be a vertex set that induces a P4 in G. A vertex v of G is said a partner of A if G[A ∪ {v}] contains at least two induced P4’s. Finally, G is called P4-tidy if each vertex set A that induces a P4 in G has at most one partner. Then, P4-tidy graphs are a superclass of cographs. + I If G is a graph, G is the graph obtained by adding a universal vertex to G.
The local complement of a graph G with respect to a vertex u ∈ V (G) is the graph G ∗ u that arises from G by replacing the induced subgraph G[NG (u)] by its complement.
Two graphs G and H are locally equivalent if and only if G arises from H by a finite sequence of local complementations.
I
Theorem. Let G be a P4-tidy graph. Then, G is a circle graph if and only if G contains no W5, net+, tent+, or tent-with-center as induced subgraph.
Theorem (Bouchet, 1994) Let G be a graph. Then, G is a circle graph if and only if no graph locally equivalent to G contains W5, W7, or BW3 as induced subgraph.
W5
BW3
W7
Edge subdivision Theorem. Let G be a graph. If G is not a circle graph, then any graph H that arises from G by edge subdivisions is not a circle graph. I
A prism is a graph that arises from C6 by a (possibly empty) sequence of subdivisions of the edges not belonging to a triangle.
I
Since C6 is locally equivalent to W5, prisms are not circle graphs.
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Tree-cographs Tree-cographs are another generalization of cographs. They are defined recursively as follows: trees are tree-cographs; the disjoint union of tree-cographs is a tree-cograph; and the complement of a tree-cograph is also a tree-cograph. It is immediate from the definition that, if G is a tree-cograph, then G or G is disconnected, or G or G is a tree. Theorem. Let G be a tree-cograph. Then, G is a circle graph if and only if G contains no induced (bipartite-claw)+ and no induced co-(bipartite-claw).
Helly circle graphs A graph is a Helly circle graph if it has a circle model whose chords are pairwise different and satisfy the Helly property (i.e., every subset of pairwise intersecting chords has a common point). I A graph is unit circle if it admits a circle model in which all the chords have the same length. I A graph is unit Helly circle if it admits a circle model in which all the chords have the same length, are pairwise different, and satisfy the Helly property. I
W5
local complement of W5
Linear Domino Graphs A graph G is domino if all its vertices belong to at most two cliques. If, in addition, each of its edges belongs to at most one clique, then G is a linear domino graph. Linear domino graphs coincide with {claw,diamond}-free graphs.
Characterizations of Helly circle graphs and unit Helly circle graphs Theorem. Let G be a graph. Then, the following assertions are equivalent: 1. G is a unit Helly circle graph. 2. G contains no induced claw, paw, diamond, or Cn∗ for any n ≥ 3. 3. G is a chordless cycle, a complete graph, or a disjoint union of chordless paths.
Theorem. Let G be a linear domino graph. Then, G is a circle graph if and only if G contains no induced prisms. {Chair,triangle}-free graphs Theorem. Let G be a {chair,triangle}-free graph. Then, G is a circle graph if and only if G contains no induced BW3. Acknowledgements Partially supported by ANPCyT PICT-2007-00518 and PICT-2007-00533, and UBACyT Grants X069 and X606 (Argentina), and FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile).
http://www.dc.uba.ar/inv/grupos/grafos
[email protected], {fbonomo,lgrippo,mdsafe}@dc.uba.ar
Permutation graphs
Circle Graphs A circle graph is the intersection graph of a family of chords on a circle. More precisely, a graph G = (V , E) is a circle graph if it is the intersection graph of a family {Lv }v ∈V of chords on a circle; i.e., v, w ∈ V are adjacent if and only if Lv ∩ Lw 6= ∅ and v 6= w . The family {Lv }v∈V is called a circle model of G.
circle model circle graph
P4-tidy graphs
Bouchet’s Characterization I
I
A graph is a comparability graph if its edges can be transitively oriented. A graph G is a permutation graph if G and G are comparability graphs. A characterization of comparability graphs by Gallai (1967) leads immediately to a forbidden induced subgraph characterization of permutation graphs. We denote by G1 + G2 the join graph of G1 and G2, where V (G1 + G2) = V (G1) ∪ V (G2) and E(G1 + G2) = E(G1) ∪ E(G2) ∪ {uv : u ∈ V (G1) and v ∈ V (G2)}. Theorem (Golumbic, 1980). The join G = G1 + G2 is a circle graph if and only if both G1 and G2 are permutation graphs. The above result is crucial in proving the following two characterizations.
Cographs are the P4-free graphs, that is, graphs with no induced path of four vertices. It is well-known that cographs are circle graphs. I Let G be a graph and let A be a vertex set that induces a P4 in G. A vertex v of G is said a partner of A if G[A ∪ {v}] contains at least two induced P4’s. Finally, G is called P4-tidy if each vertex set A that induces a P4 in G has at most one partner. Then, P4-tidy graphs are a superclass of cographs. + I If G is a graph, G is the graph obtained by adding a universal vertex to G.
The local complement of a graph G with respect to a vertex u ∈ V (G) is the graph G ∗ u that arises from G by replacing the induced subgraph G[NG (u)] by its complement.
Two graphs G and H are locally equivalent if and only if G arises from H by a finite sequence of local complementations.
I
Theorem. Let G be a P4-tidy graph. Then, G is a circle graph if and only if G contains no W5, net+, tent+, or tent-with-center as induced subgraph.
Theorem (Bouchet, 1994) Let G be a graph. Then, G is a circle graph if and only if no graph locally equivalent to G contains W5, W7, or BW3 as induced subgraph.
W5
BW3
W7
Edge subdivision Theorem. Let G be a graph. If G is not a circle graph, then any graph H that arises from G by edge subdivisions is not a circle graph. I
A prism is a graph that arises from C6 by a (possibly empty) sequence of subdivisions of the edges not belonging to a triangle.
I
Since C6 is locally equivalent to W5, prisms are not circle graphs.
=
Tree-cographs Tree-cographs are another generalization of cographs. They are defined recursively as follows: trees are tree-cographs; the disjoint union of tree-cographs is a tree-cograph; and the complement of a tree-cograph is also a tree-cograph. It is immediate from the definition that, if G is a tree-cograph, then G or G is disconnected, or G or G is a tree. Theorem. Let G be a tree-cograph. Then, G is a circle graph if and only if G contains no induced (bipartite-claw)+ and no induced co-(bipartite-claw).
Helly circle graphs A graph is a Helly circle graph if it has a circle model whose chords are pairwise different and satisfy the Helly property (i.e., every subset of pairwise intersecting chords has a common point). I A graph is unit circle if it admits a circle model in which all the chords have the same length. I A graph is unit Helly circle if it admits a circle model in which all the chords have the same length, are pairwise different, and satisfy the Helly property. I
W5
local complement of W5
Linear Domino Graphs A graph G is domino if all its vertices belong to at most two cliques. If, in addition, each of its edges belongs to at most one clique, then G is a linear domino graph. Linear domino graphs coincide with {claw,diamond}-free graphs.
Characterizations of Helly circle graphs and unit Helly circle graphs Theorem. Let G be a graph. Then, the following assertions are equivalent: 1. G is a unit Helly circle graph. 2. G contains no induced claw, paw, diamond, or Cn∗ for any n ≥ 3. 3. G is a chordless cycle, a complete graph, or a disjoint union of chordless paths.
Theorem. Let G be a linear domino graph. Then, G is a circle graph if and only if G contains no induced prisms. {Chair,triangle}-free graphs Theorem. Let G be a {chair,triangle}-free graph. Then, G is a circle graph if and only if G contains no induced BW3. Acknowledgements Partially supported by ANPCyT PICT-2007-00518 and PICT-2007-00533, and UBACyT Grants X069 and X606 (Argentina), and FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile).
http://www.dc.uba.ar/inv/grupos/grafos
[email protected], {fbonomo,lgrippo,mdsafe}@dc.uba.ar
