Weighted connected domination and Steiner trees ... - Semantic Scholar
DISCRETE APPLIED MATHEMATICS ELSEVIER
Discrete Applied Mathematics
87 (1998) 245-253
Weighted connected domination and Steiner trees in distance-hereditary graphs * Hong-Gwa
Yeh, Gerard J. Chang”
Depurtment of Applied Mathematics, Nationul Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan Received
20 September
1994 received in revised form 3 March 1998; accepted
9 March 1998
Abstract Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph. Both problems are MY-complete in general graphs. 0 1998 Elsevier Science B.V. All rights reserved.
Keywords: Distance-hereditary
graph; Connected
domination;
Steiner tree; Algorithm;
Cograph
1. Introduction The concept erations
of domination
research.
can be used to model
many location
problems
in op-
In a graph G = (V,E), a dominating set is a subset D of vertices
such that every vertex in V - D is adjacent of G is connected if the subgraph
to some vertex in D. A dominating set G[D] induced by D is connected. The connected
domination problem is to find a minimum-sized Suppose,
moreover,
connected
that each vertex u in G is associated
dominating
set of a graph.
with a weight W(V) that is
a real number. The weighted connected domination problem is to find a connected dominating set D such that w(D) = COEDw(v) is as small as possible. The concept is also closely vertices
of Steiner trees originally related to connected
concerned
domination
points in Euclidean
in graphs.
Suppose
spaces, but it
T is a subset of
in a graph G = (V, E). The Steiner tree problem is to find a minimal
subset
S of V - T such that G[S U T] is connected. S and T are called the Steiner set and target set, respectively. We can also consider the vertex-weighted version of the Steiner tree problem, which was originally introduced by Segev [27]. The vertex weight of a * Supported in part by the National Science Council under grant NSC84-2121-M009-023. * Corresponding
author. E-mail: [email protected].
0166-218X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI2 SO 166-2 18X(98 )00060-2
246
H.-G. Yeh, G.J. Changi Discrete Applied Mathematics 87 (1998)
Steiner vertex can be interpreted tree. Traditionally,
the problem
as the cost of adding this vertex when forming
all complete
tree problem is _&Y-complete contain
graphs (see [21]). However,
out that the edge-weighted
any classes that contains the edge-weighted
Steiner tree problem The connected
complete
graphs
[18, 251, split graphs
[7], cographs
bipartite
graphs [24]. It is also known that the connected for k-trees
~-CUBS
(k 22)
geneous
graphs [ 141.
[ 1 I]. The Steiner
versions.
or objective
Results
usual version. weighted chordal
graphs
graphs
the corresponding
for these variant
[3]. The purpose
graphs
domination
domination
problems
problem
domination problems
independent
perfect
problem
is poly-
solvable
problems
for
in homo-
may have differ-
are r-domination
and weighted
are relatively
fewer than the
domination
domination
for
[23, 301, and chordal
results of this kind are polynomial
[ 171, the weighted
r-domination
graphs
for
for strongly
[9, 221, series-parallel
is polynomially
Typical examples
and the weighted
graphs [6], the r-domination connected
[23, 301, chordal
tree problem
functions.
Some well-known
domination
solvable
(fixed k) [l] and ~-CUBS [l l] and MY-complete
For many location problems, ent constraints
have the same complexity
graphs [3, 131; and they are MY-complete
graphs solvable
Steiner
graphs as these
the vertex-weighted
they are both polynomially
graphs
for
graphs in this paper.
bipartite nomially
the edge-weighted
distance-hereditary
and Steiner tree problems
[12, 26, 29, 301, and distance-hereditary
[22, p. 445,
is NP-complete
graphs. So, we only consider
for distance-hereditary
[30], permutation
Johnson
tree problem
graphs. In particular,
many classes of graphs. For instance, chordal
Steiner
for the edge-weighted
domination
the
of finding the Steiner tree for a set of points in a graph
has been studied for edge-weighted line 91, pointed
245-253
algorithms
problems
problem
for the
in strongly
in co-comparability
in trees [28] and strongly chordal graphs [5], the
in strongly chordal graphs [S] and distance-hereditary
of this paper
is to present
linear-time
algorithms
for the
weighted connected domination problem with arbitrary weights and the vertex-weighted Steiner tree problem with non-negative weights in distance-hereditary graphs. In the rest of this section,
we give a brief survey of distance-hereditary
graph is distance-hereditary
if every two vertices
connected-induced
Distance-hereditary
subgraph.
[20]. The characterization
and recognition
graphs were introduced
of distance-hereditary
ied in [2, 13, 15, 19, 201. Note that the class of distance-hereditary of all parity graphs [4] and a superclass
graphs. A
have the same distance
of all cographs
in every
by Howorka
graphs have been studgraphs is a subclass
[8, lo].
Suppose A and B are two sets of vertices in a graph G = (V,E). The neighborhood No of B in A is the set of vertices in A that are adjacent to some vertex in B. The closed neighborhood &[B] of B in A is NA[B]UB. For simplicity, NA(u), NA[u], N(B), and NW stand for NA({D}),N~[{u}l, NO>, and Nv[B], respectively. The distance dC(x, y) or d(x, y) between two vertices x and y in G is the minimum length of an x-y path in G. The hanging h, of a connected graph G = (V, E) at a vertex 11E V is the collection of sets LO(U), L,(u), . . . , L,(u) (or Lo, Li,. . ,Lt if there is no ambiguity), where t = maxVEvdo(u,v) and Li(U) = {u E V : d~(u,v) = i} for O
w’(D), i.e., w(M - V’) = w’(M)>w’(D)
= w(D - V’), and so
w(M)=w(M-V’)+w(MflV’)~w(D-V’)+w(V’)=w(DuV’). This completes
the proof of the lemma.
0
Lemma 4 suggests that it suffices to consider problem with a non-negative weight function.
the weighted
connected
domination
248
H.-G. Yeh, G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253
Lemma 5. Suppose h, = {Lo,Ll,. . ., L,} is a hanging of a connected distancehereditary graph at u. For any connected dominating set D and v E Li with 2 IDI. In this case, there is at least one vertex x in M that is not a
ii++. So
This together with Claim 2 proves dominating set of G.
that D U {u}
is a minimum
weighted
connected
H.-G.
Yeh. G.J. Changl Discrete
Case 3: IMI = IDI 22.
Since d
Applied Mathematics
contains
pairwise
249
87 (1998) 245-253
disjoint
sets, M = {u**: N’(u) E
d}. so w(M) = c,** w(u**)> c,. w(v*) = w(D). For any two vertices x* and y* in D, x** and y** are in M. Since G[M] is connected, there is an x**-y** x
**
**
=v
0
2
** v,
For any 1 di,_,)(u,*_~). This proves to VT for 1
Discrete Applied Mathematics
87 (1998) 245-253
Weighted connected domination and Steiner trees in distance-hereditary graphs * Hong-Gwa
Yeh, Gerard J. Chang”
Depurtment of Applied Mathematics, Nationul Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan Received
20 September
1994 received in revised form 3 March 1998; accepted
9 March 1998
Abstract Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph. Both problems are MY-complete in general graphs. 0 1998 Elsevier Science B.V. All rights reserved.
Keywords: Distance-hereditary
graph; Connected
domination;
Steiner tree; Algorithm;
Cograph
1. Introduction The concept erations
of domination
research.
can be used to model
many location
problems
in op-
In a graph G = (V,E), a dominating set is a subset D of vertices
such that every vertex in V - D is adjacent of G is connected if the subgraph
to some vertex in D. A dominating set G[D] induced by D is connected. The connected
domination problem is to find a minimum-sized Suppose,
moreover,
connected
that each vertex u in G is associated
dominating
set of a graph.
with a weight W(V) that is
a real number. The weighted connected domination problem is to find a connected dominating set D such that w(D) = COEDw(v) is as small as possible. The concept is also closely vertices
of Steiner trees originally related to connected
concerned
domination
points in Euclidean
in graphs.
Suppose
spaces, but it
T is a subset of
in a graph G = (V, E). The Steiner tree problem is to find a minimal
subset
S of V - T such that G[S U T] is connected. S and T are called the Steiner set and target set, respectively. We can also consider the vertex-weighted version of the Steiner tree problem, which was originally introduced by Segev [27]. The vertex weight of a * Supported in part by the National Science Council under grant NSC84-2121-M009-023. * Corresponding
author. E-mail: [email protected].
0166-218X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved. PI2 SO 166-2 18X(98 )00060-2
246
H.-G. Yeh, G.J. Changi Discrete Applied Mathematics 87 (1998)
Steiner vertex can be interpreted tree. Traditionally,
the problem
as the cost of adding this vertex when forming
all complete
tree problem is _&Y-complete contain
graphs (see [21]). However,
out that the edge-weighted
any classes that contains the edge-weighted
Steiner tree problem The connected
complete
graphs
[18, 251, split graphs
[7], cographs
bipartite
graphs [24]. It is also known that the connected for k-trees
~-CUBS
(k 22)
geneous
graphs [ 141.
[ 1 I]. The Steiner
versions.
or objective
Results
usual version. weighted chordal
graphs
graphs
the corresponding
for these variant
[3]. The purpose
graphs
domination
domination
problems
problem
domination problems
independent
perfect
problem
is poly-
solvable
problems
for
in homo-
may have differ-
are r-domination
and weighted
are relatively
fewer than the
domination
domination
for
[23, 301, and chordal
results of this kind are polynomial
[ 171, the weighted
r-domination
graphs
for
for strongly
[9, 221, series-parallel
is polynomially
Typical examples
and the weighted
graphs [6], the r-domination connected
[23, 301, chordal
tree problem
functions.
Some well-known
domination
solvable
(fixed k) [l] and ~-CUBS [l l] and MY-complete
For many location problems, ent constraints
have the same complexity
graphs [3, 131; and they are MY-complete
graphs solvable
Steiner
graphs as these
the vertex-weighted
they are both polynomially
graphs
for
graphs in this paper.
bipartite nomially
the edge-weighted
distance-hereditary
and Steiner tree problems
[12, 26, 29, 301, and distance-hereditary
[22, p. 445,
is NP-complete
graphs. So, we only consider
for distance-hereditary
[30], permutation
Johnson
tree problem
graphs. In particular,
many classes of graphs. For instance, chordal
Steiner
for the edge-weighted
domination
the
of finding the Steiner tree for a set of points in a graph
has been studied for edge-weighted line 91, pointed
245-253
algorithms
problems
problem
for the
in strongly
in co-comparability
in trees [28] and strongly chordal graphs [5], the
in strongly chordal graphs [S] and distance-hereditary
of this paper
is to present
linear-time
algorithms
for the
weighted connected domination problem with arbitrary weights and the vertex-weighted Steiner tree problem with non-negative weights in distance-hereditary graphs. In the rest of this section,
we give a brief survey of distance-hereditary
graph is distance-hereditary
if every two vertices
connected-induced
Distance-hereditary
subgraph.
[20]. The characterization
and recognition
graphs were introduced
of distance-hereditary
ied in [2, 13, 15, 19, 201. Note that the class of distance-hereditary of all parity graphs [4] and a superclass
graphs. A
have the same distance
of all cographs
in every
by Howorka
graphs have been studgraphs is a subclass
[8, lo].
Suppose A and B are two sets of vertices in a graph G = (V,E). The neighborhood No of B in A is the set of vertices in A that are adjacent to some vertex in B. The closed neighborhood &[B] of B in A is NA[B]UB. For simplicity, NA(u), NA[u], N(B), and NW stand for NA({D}),N~[{u}l, NO>, and Nv[B], respectively. The distance dC(x, y) or d(x, y) between two vertices x and y in G is the minimum length of an x-y path in G. The hanging h, of a connected graph G = (V, E) at a vertex 11E V is the collection of sets LO(U), L,(u), . . . , L,(u) (or Lo, Li,. . ,Lt if there is no ambiguity), where t = maxVEvdo(u,v) and Li(U) = {u E V : d~(u,v) = i} for O
w’(D), i.e., w(M - V’) = w’(M)>w’(D)
= w(D - V’), and so
w(M)=w(M-V’)+w(MflV’)~w(D-V’)+w(V’)=w(DuV’). This completes
the proof of the lemma.
0
Lemma 4 suggests that it suffices to consider problem with a non-negative weight function.
the weighted
connected
domination
248
H.-G. Yeh, G.J. Changl Discrete Applied Mathematics 87 (1998) 245-253
Lemma 5. Suppose h, = {Lo,Ll,. . ., L,} is a hanging of a connected distancehereditary graph at u. For any connected dominating set D and v E Li with 2 IDI. In this case, there is at least one vertex x in M that is not a
ii++. So
This together with Claim 2 proves dominating set of G.
that D U {u}
is a minimum
weighted
connected
H.-G.
Yeh. G.J. Changl Discrete
Case 3: IMI = IDI 22.
Since d
Applied Mathematics
contains
pairwise
249
87 (1998) 245-253
disjoint
sets, M = {u**: N’(u) E
d}. so w(M) = c,** w(u**)> c,. w(v*) = w(D). For any two vertices x* and y* in D, x** and y** are in M. Since G[M] is connected, there is an x**-y** x
**
**
=v
0
2
** v,
For any 1 di,_,)(u,*_~). This proves to VT for 1
