Paired-dominating Set Research Articles

Well paired-dominated graphs

A paired-dominating set is a dominating set whose induced subgraph contains at least one perfect matching. This could model the situation of guards or police where each has a partner or backup. We are interested in those where all “minimal” paired-dominating sets are the same cardinality. In this case, we consider “minimal” to be with respect to the pairings. That is, the removal of any two vertices paired under the matching results in a set that is not dominating. We give a structural characterization of all such graphs with girth at least eight.

The diameter of paired-domination vertex critical graphs

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 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 graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ pr(G − v) < γ pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ pr(G) − 2). For γ pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph.

Paired-Domination in P 5-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 $$\gamma_pr(G)$$, is the minimum cardinality of a paired-dominating set of G. In [1], 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 P 5, which seems to be the critical case.

Labelling algorithms for paired-domination problems in block and interval graphs

Let G=(V,E) be a graph without isolated vertices. A set S⊆V is a paired-dominating set if every vertex in V−S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang and Ng (Discrete Appl. Math. 155:2077–2086, 2007), we present two linear time algorithms to find a minimum cardinality paired-dominating set in block and interval graphs. In addition, we prove that paired-domination problem is NP-complete for bipartite graphs, chordal graphs, even for split graphs.

A polynomial-time algorithm for the paired-domination problem on permutation graphs

A set S of vertices in a graph H = ( V , E ) with no isolated vertices is a paired-dominating set of H if every vertex of H is adjacent to at least one vertex in S and if the subgraph induced by S contains a perfect matching. Let G be a permutation graph and π be its corresponding permutation. In this paper we present an O ( m n ) time algorithm for finding a minimum cardinality paired-dominating set for a permutation graph G with n vertices and m edges.

Upper bounds on the paired-domination number

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 the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G , denoted by γ p r ( G ) . In this work, we present several upper bounds on the paired-domination number in terms of the maximum degree, minimum degree, girth and order.

Paired-domination number of a graph and its complement

A paired-dominating set of a graph G = ( V , E ) with no isolated vertex is a dominating set of vertices inducing a graph with a perfect matching. The paired-domination number of G , denoted by γ p r ( G ) , is the minimum cardinality of a paired-dominating set of G . We consider graphs of order n ≥ 6 , minimum degree δ such that G and G ¯ do not have an isolated vertex and we prove that –if γ p r ( G ) > 4 and γ p r ( G ¯ ) > 4 , then γ p r ( G ) + γ p r ( G ¯ ) ≤ 3 + min { δ ( G ) , δ ( G ¯ ) } . –if δ ( G ) ≥ 2 and δ ( G ¯ ) ≥ 2 , then γ p r ( G ) + γ p r ( G ¯ ) ≤ 2 n 3 + 4 and γ p r ( G ) + γ p r ( G ¯ ) ≤ 2 n 3 + 2 if moreover n ≥ 21 .

A characterization of [formula omitted]-trees

Let G = ( V , E ) be a graph. A set S ⊆ V is a dominating set of G if every vertex not in S is adjacent with some vertex in S. The domination number of G, denoted by γ ( G ) , is the minimum cardinality of a dominating set of G. A set S ⊆ V is a paired-dominating set of G if S dominates V and 〈 S 〉 contains at least one perfect matching. The paired-domination number of G, denoted by γ p ( G ) , is the minimum cardinality of a paired-dominating set of G. In this paper, we provide a constructive characterization of those trees for which the paired-domination number is twice the domination number.

Vertices contained in all minimum paired-dominating sets of a tree

A set S of vertices in a graph G is called a paired-dominating set if it dominates V and 〈S〉 contains at least one perfect matching. We characterize the set of vertices of a tree that are contained in all minimum paired-dominating sets of the tree.

Upper total domination versus upper paired-domination

Let G be a graph with no isolated vertices. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S, while a paired-dominating set of G is a dominating set of vertices whose induced subgraph has a perfect matching. The maximum cardinality of a minimal total dominating set and a minimal paired-dominating set of G is the upper total domination number and upper paired-domination number of G, respectively, denoted by Γ t (G) and Γpr(G). In this paper, we investigate the relationship between the upper total domination and upper paired-domination numbers of a graph. We show that for every graph G with no isolated vertex Γ t (G) ≥ 1/2(Γpr(G) + 2), and we characterize the trees achieving this bound. For each positive integer k, we observe that there exist connected graphs G and H such that Γpr(G) – Γ t (G) ≥ k and Γ t (H) – Γpr(H) ≥ k.However for the family of trees Ton at least two vertices, we show that Γ t (T) ≤ Γ t (T).

Paired-domination in generalized claw-free graphs

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). 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 \(\gamma_{\rm pr}(G)\), is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order \(n \ge 3\), then \(\gamma_{\rm pr}(G) \le n-1\) and this bound is sharp for graphs of arbitrarily large order. Every graph is \(K_{1,a+2}\)-free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected \(K_{1,a+2}\)-free graph of order n ≥ 2, then \(\gamma_{\rm pr}(G) \le 2(an + 1)/(2a+1)\) with infinitely many extremal graphs.

Graphs with large paired-domination number

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (1998) Networks 32: 199–206. A paired-dominating set of a graph G 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 \(\gamma_{\rm pr}(G)\), is the minimum cardinality of a paired-dominating set of G. Let G be a connected graph of order n with minimum degree at least two. Haynes and Slater (1998) Networks 32: 199–206, showed that if n ≥ 6, then \(\gamma_{\rm pr}(G) \le 2n/3\). In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that \(\gamma_{\rm pr}(G) \le 2(n-1)/3\) and we characterize the (infinite family of) graphs that achieve equality in this bound.

Characterizations of trees with equal paired and double domination numbers

A paired-dominating set of a graph G is a dominating set of vertices whose induced subgraph has a perfect matching, and a double dominating set is a dominating set that dominates every vertex of G at least twice. We show that for trees, the paired-domination number is less than or equal to the double domination number, solving a conjecture of Chellali and Haynes. Then we characterize the trees having equal paired and double domination numbers.

Vertices Contained in all or in no Minimum Paired-Dominating Set of a Tree

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. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree.

Paired-Domination in Claw-Free Cubic 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 minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by γpr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K1,3 or K4−e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K1,3,K4−e,C4)-free, then γpr(G)≤3n/8; (ii) if G is claw-free and diamond-free, then γpr(G)≤2n/5; (iii) if G is claw-free, then γpr(G)≤n/2. In all three cases, the extremal graphs are characterized.

Paired-domination in inflated graphs

The inflation G I of a graph G with n ( G ) vertices and m ( G ) edges is obtained from G by replacing every vertex of degree d of G by a clique K d . 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 γ p ( G ) is the minimum cardinality of a paired dominating set of G . In this paper, we show that if a graph G has a minimum degree δ ( G )⩾2, then n ( G )⩽ γ p ( G I )⩽4 m ( G )/[ δ ( G )+1], and the equality γ p (G I ) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.

Paired-domination in inflated graphs

The inflation G I of a graph G with n( G) vertices and m( G) edges is obtained from G by replacing every vertex of degree d of G by a clique K d . 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 γ p( G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ( G)⩾2, then n( G)⩽ γ p ( G I )⩽4 m( G)/[ δ( G)+1], and the equality γ p(G I) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.

Total and paired-domination numbers of a tree

A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S. A paired-dominating set of G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired-dominating set) is the total domination number γt(G) (respectively, the paired-domination number γpr(G)). We give sharp upper bounds on the total and paired-domination numbers of trees that improve known bounds for some cases. in particular, we show that for a tree T with order n ≥ 3 and s support vertices, γpr(T)  γt(T) + s — 1, γt(T) ≤ (n + s)/2, and γpr(T) ≤ (n + 2s — 1)/2.

Paired-domination of Trees

Let G= (V, E) be a graph without isolated vertices. A set S⊂V is a paired-dominating set if it dominates V and the subgraph induced by S,≤S\ge, contains a perfect matching. The paired-domination number γp(G) is defined to be the minimum cardinality of a paired-dominating set S in G. In this paper, we present a linear-time algorithm computing the paired-domination number for trees and characterize trees with equal domination and paired-domination numbers.