Paired-dominating Set Research Articles

Paired versus double domination in K 1,r -free graphs

A vertex in G is said to dominate itself and its neighbors. A subset S of vertices is a dominating set if S dominates every vertex of G. A paired-dominating set is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by ? pr(G), is the minimum cardinality of a paired-dominating set in G. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number ? ×2(G). A claw-free graph is a graph that does not contain K 1,3 as an induced subgraph. Chellali and Haynes (Util. Math. 67:161---171, 2005) showed that for every claw-free graph G, we have ? pr(G)≤? ×2(G). In this paper we extend this result by showing that for r?2, if G is a connected graph that does not contain K 1,r as an induced subgraph, then $\gamma_{\mathrm{pr}}(G)\le ( \frac{2r^{2}-6r+6}{r(r-1)} )\gamma_{\times2}(G)$ .

Paired-Domination in Claw-Free Graphs

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number, denoted by ? pr (G), is the minimum cardinality of a paired-dominating set in G. In this paper we investigate the paired-domination number in claw-free graphs. Specifically, we show that ? pr (G) ≤ (3n ? 1)/5 if G is a connected claw-free graph of order n with minimum degree at least three and that this bound is sharp.

A linear time algorithm for computing a minimum paired-dominating set of a convex bipartite graph

A set D of vertices of a graph G=(V,E) is a dominating set of G if every vertex in V∖D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if the induced subgraph, G[D], has a perfect matching. The paired-domination problem is for a given graph G and a positive integer k to answer if G has a paired-dominating set of size at most k. The paired-domination problem is known to be NP-complete even for bipartite graphs. In this paper, we propose a linear time algorithm to compute a minimum paired-dominating set of a convex bipartite graph.

Minimum paired-dominating set in chordal bipartite graphs and perfect elimination bipartite graphs

A set D⊆V of a graph G=(V,E) is a dominating set of G if every vertex in V?D has at least one neighbor in D. A dominating set D of G is a paired-dominating set of G if the induced subgraph, G[D], has a perfect matching. Given a graph G=(V,E) and a positive integer k, the paired-domination problem is to decide whether G has a paired-dominating set of cardinality at most k. The paired-domination problem is known to be NP-complete for bipartite graphs. In this paper, we, first, strengthen this complexity result by showing that the paired-domination problem is NP-complete for perfect elimination bipartite graphs. We, then, propose a linear time algorithm to compute a minimum paired-dominating set of a chordal bipartite graph, a well studied subclass of bipartite graphs.

Paired- and induced paired-domination in (E,net)-free graphs

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by attaching a pendant vertex to each vertex of the path on three vertices) as induced subgraphs. This graph class is a natural generalization of {claw, net}-free graphs, which are intensively studied with respect to their nice properties concerning domination and hamiltonicity. We show that any connected {E, net}-free graph has a paired-dominating set that, roughly, contains at most half of the vertices of the graph. This bound is a significant improvement to the known general bounds. Further, we show that any {E, net, C5}-free graph has an induced paireddominating set, that is a paired-dominating set that forms an induced matching, and that such set can be chosen to be a minimum paired-dominating set. We use these results to obtain a new characterization of {E, net, C5}-free graphs in terms of the hereditary existence of induced paired-dominating sets. Finally, we show that the induced matching formed by an induced paired-dominating set in a {E, net, C5}-free graph can be chosen to have at most two times the size of the smallest maximal induced matching possible.

Linear-time algorithm for the matched-domination problem in cographs

Let G=(V, E) be a graph without isolated vertices. A matching in G is a set of independent edges in G. A perfect matching M in G is a matching such that every vertex of G is incident to an edge of M. A set S⊆ V is a paired-dominating set of G if every vertex not in S is adjacent to a vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination problem is to find a paired-dominating set of G with minimum cardinality. This paper introduces a generalization of the paired-domination problem, namely, the matched-domination problem, in which some constrained vertices are in paired-dominating sets as far as they can. Further, possible applications are also presented. We then present a linear-time constructive algorithm to solve the matched-domination problem in cographs.† †A preliminary version of this paper has appeared in: Proceedings of the 4th IASTED International Conference on Computational Intelligence (CI’2009), Honolulu, Hawaii, USA, 2009, pp. 120–126.

On the existence of total dominating subgraphs with a prescribed additive hereditary property

Recently, Bacsó and Tuza gave a full characterization of the graphs for which every connected induced subgraph has a connected dominating subgraph satisfying an arbitrary prescribed hereditary property. Using their result, we derive a similar characterization of the graphs for which any isolate-free induced subgraph has a total dominating subgraph that satisfies a prescribed additive hereditary property. In particular, we give a characterization for the case where the total dominating subgraphs are a disjoint union of complete graphs. This yields a characterization of the graphs for which every isolate-free induced subgraph has a vertex-dominating induced matching, a so-called induced paired-dominating set.

Vertices in all minimum paired-dominating sets of block graphs

Let G=(V,E) be a simple graph without isolated vertices. A set S?V is a paired-dominating set if every vertex in V?S has at least one neighbor in S and the subgraph induced by S contains a perfect matching. In this paper, we present a linear-time algorithm to determine whether a given vertex in a block graph is contained in all its minimum paired-dominating sets.

An upper bound on the paired-domination number in terms of the number of edges in the graph

In this paper, we continue the study of paired domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998) 199–206]. A paired-dominating set of a graph is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G , denoted by γ pr ( G ) , is the minimum cardinality of a paired-dominating set in G . We show that if G is a connected graph of size m ≥ 18 with minimum degree at least 2, then γ pr ( G ) ≤ 4 m / 7 and we characterize the (infinite family of) graphs that achieve equality in this bound.

Graphs with disjoint dominating and paired-dominating sets

Abstract A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating set and a paired-dominating set.

Paired-Domination in Subdivided Star-Free Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γ pr(G), is the minimum cardinality of a paired-dominating set of G. In [Dorbec P, Gravier S, Henning MA, J Comb Optim 14(1):1–7, 2007], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced subdivided stars, depending on the size of the star.

Upper paired-domination in claw-free graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γpr(G). We establish bounds on Γpr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree ?. We show that Γpr(G)?4n/5 if ?=1 and n?3, Γpr(G)?3n/4 if ?=2 and n?6, and Γpr(G)?2n/3 if ??3. All these bounds are sharp. Further, if n?6 the graphs G achieving the bound Γpr(G)=4n/5 are characterized, while for n?9 the graphs G with ?=2 achieving the bound Γpr(G)=3n/4 are characterized.

A Characterization of Cubic Graphs with Paired-Domination Number Three-Fifths Their Order

A paired-dominating set of a graph is a dominating set of vertices whose induced subgraph has a perfect matching, while the paired-domination number is the minimum cardinality of a paired-dominating set in the graph. Recently, Chen et al. (Acta Math Sci Ser A Chin Ed 27(1):166–170, 2007) proved that a cubic graph has paired-domination number at most three-fifths the number of vertices in the graph. In this paper, we show that the Petersen graph is the only connected cubic graph with paired-domination number three-fifths its order.

Paired-Domination Subdivision Numbers of Graphs

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ pr (G), is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number sd γpr (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. In this paper we establish upper bounds on the paired-domination subdivision number and pose some problems and conjectures.

A characterization of graphs with disjoint dominating and paired-dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set and the subgraph induced by the set contains a perfect matching. In this paper, we provide a constructive characterization of graphs whose vertex set can be partitioned into a dominating set and a paired-dominating set.

A linear-time algorithm for paired-domination problem in strongly chordal graphs

Let G = ( V , E ) be a simple graph without isolated vertices. A vertex set S ⊆ V is a paired-dominating set if every vertex in V − S has at least one neighbor in S and the induced subgraph G [ S ] has a perfect matching. In this paper, we present a linear-time algorithm to find a minimum paired-dominating set in strongly chordal graphs if the strong (elimination) ordering of the graph is given in advance.

Which trees have a differentiating-paired dominating set?

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. The set S is called a differentiating-paired dominating set if for every pair of distinct vertices u and v in V(G), N[u]∩S≠N[v]∩S, where N[u] denotes the set consisting of u and all vertices adjacent to u. In this paper, we provide a constructive characterization of trees that do not have a differentiating-paired dominating set.

Hardness results and approximation algorithms for (weighted) paired-domination in graphs

Let G = ( V , E ) be a simple graph without isolated vertices. A vertex set S ⊆ V is a paired-dominating set if every vertex in V − S has a neighbor in S and the induced subgraph G [ S ] has a perfect matching. In this paper, we investigate the approximation hardness of paired-domination in graphs. For weighted paired-domination, an approximation algorithm in general graphs and an exact dynamic programming style algorithm in trees are also given.

Distance paired-domination problems on subclasses of chordal graphs

Let G = ( V , E ) be a graph without isolated vertices. For a positive integer k , a set S ⊆ V is a k -distance paired-dominating set if each vertex in V − S is within distance k of a vertex in S and the subgraph induced by S contains a perfect matching. In this paper, we present two linear time algorithms to find a minimum cardinality k -distance paired-dominating set in interval graphs and block graphs, which are two subclasses of chordal graphs. In addition, we present a characterization of trees with unique minimum k -distance paired-dominating set.

Locating and paired-dominating sets in graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater [T.W. Haynes, P.J. Slater, Paired-domination in graphs, Networks 32 (1998), 199–206]. A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. We consider paired-dominating sets which are also locating sets, that is distinct vertices of G are dominated by distinct subsets of the paired-dominating set. We consider three variations of sets which are paired-dominating and locating sets and investigate their properties.