Disjunctive Domination Research Articles

Some results on disjunctive domination in graphs

Some results on disjunctive domination in graphs

Algorithmic aspects of paired disjunctive domination in graphs

In a graph G=(V,E) without an isolated vertex, a dominating set D⊆V is a paired dominating set if the graph G[D] induced by D has a perfect matching. Further, a set D⊆V is a disjunctive dominating set of G if for each vertex v∈V, either NG[v]∩D≠∅ or there are at least two vertices in D whose distance from v is two in G. We introduce the notion of paired disjunctive domination in graphs. A disjunctive dominating set D⊆V in the graph G is a paired disjunctive dominating set if G[D] has a perfect matching. The minimum cardinality of a paired disjunctive dominating set of G is the paired disjunctive domination number, denoted by γprd(G).In this article, we compute the exact value of γprd(G) when G is a path, cycle, cograph, chain graph, and split graph. We prove that the decision version of the problem is NP-complete for planar graphs, bipartite graphs, and chordal graphs and design a polynomial-time algorithm to compute a minimum cardinality paired disjunctive dominating set in interval graphs. Further, we obtain lower and upper bounds on the approximation ratio of the problem and proved that the problem is APX-complete for the graphs with maximum degree 4.

Disjunctive domination in graphs with minimum degree at least two

A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ2d(G), of G is the minimum cardinality of a disjunctive dominating set in G. In this paper, we show that if G be a graph of order at least 3, δ(G)≥2 and with no component isomorphic to any of eight forbidden graphs, then γ2d(G)≤|G|3. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that G is a claw-free graph with minimum degree at least two.

Bounds on the disjunctive domination number of a tree

A set D of vertices in a graph G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γd2(G), of G is the minimum cardinality of a disjunctive dominating set in G. We show that if T is a tree of order n with l leaves and s support vertices, then n-l+3/4≤γd2(T)≤n+l+s/4. Moreover, we characterize the families of trees which attain these bounds.

Restrained Disjunctive Domination in Graphs under Some Binary Operations

A set S ⊆ V (G) is a disjunctive dominating set of a graph G if for every v ∈ V (G)\S, v is a neighbor of a vertex in S or S has at least two vertices each at distance 2 from v. We say that a disjunctive dominating set S of G is a restrained disjunctive dominating set if for each v ∈ V (G)\S there exists u ∈ V (G) \ S such that uv ∈ E(G) or there exist distinct vertices u, w ∈ V (G) \ S such that dG(u, v) = 2 = dG(w, v). The minimum cardinality γdr(G) of a restrained disjunctive dominating set of G is the restrained disjunctive domination number of G. In this paper, we characterize the restrained disjunctive dominating sets in some binary operations such as the join, corona and lexicographic product of graphs and, as a result, obtain the values of their corresponding restrained disjunctive domination numbers.

On disjunctive domination number of corona related graphs

A disjunctive dominating set in a graph G is a set S ⊂ V(G) such that every vertex v ∈V(G) S is either adjacent to a vertex in S or has at least 2 vertices in S at a distance 2 from it in G. The disjunctive domination number of G, denoted by γ 2 d(G), is the minimum cardinality of a disjunctive dominating set in G. In this paper we investigate disjunctive domination number of some corona related graphs.

On disjunctive domination number of corona related graphs

A disjunctive dominating set in a graph G is a set S ⊂ V(G) such that every vertex v ∈V(G) \ S is either adjacent to a vertex in S or has at least 2 vertices in S at a distance 2 from it in G. The disjunctive domination number of G, denoted by γ2d(G), is the minimum cardinality of a disjunctive dominating set in G. In this paper we investigate disjunctive domination number of some corona related graphs.

Properties of disjunctive domination in product graphs

Properties of disjunctive domination in product graphs

On disjunctive domination in graphs

A set S of vertices of a graph G is a disjunctive dominating set if for every v ∈ V (G) \\ S, v is adjacent to a vertex in S or S contains two vertices each of distance two from v. We say that S is a disjunctive total dominating set if for every v ∈ V (G), v is adjacent to a vertex in S or S contains two vertices each of distance two from v. In this paper, we characterize the disjunctive dominating sets and the disjunctive total dominating sets in the join, corona and composition of graphs and, as a result, obtain the values of their respective corresponding disjunctive domination numbers.

Algorithmic aspects of b-disjunctive domination in graphs

For a graph \(G=(V,E)\), a set \(D\subseteq V\) is called a disjunctive dominating set of G if for every vertex \(v\in V\setminus D\), v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by \(\gamma _{2}^{d}(G)\). The Minimum Disjunctive Domination Problem (MDDP) is to find a disjunctive dominating set of cardinality \(\gamma _{2}^{d}(G)\). Given a positive integer k and a graph G, the Disjunctive Domination Decision Problem (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a polynomial time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln (\Delta ^{2}+\Delta +2)+1)\)-approximation algorithm for MDDP, where \(\Delta \) is the maximum degree of input graph \(G=(V,E)\) and prove that MDDP can not be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.

A constructive characterization of trees with equal total domination and disjunctive domination numbers

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S. The total domination number, γt(G), of G is the minimum cardinality of a total dominating set of G. A set S of vertices in G is a disjunctive dominating set in G if every vertex not in S is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it in G. The disjunctive domination number, (G), of G is the minimum cardinality of a disjunctive dominating set in G. By definition, we have (T )≤γt (T ). In this paper, we provide a constructive characterization of the trees T achieving equality in this bound.

Domination versus disjunctive domination in graphs

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number of G is the minimum cardinality of a dominating set of G. For a positive integer b, a set S of vertices in a graph G is a b-disjunctive dominating set in G if every vertex v not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it in G. The b-disjunctive domination number of G is the minimum cardinality of a b-disjunctive dominating set. In this paper, we continue the study of disjunctive domination in graphs. We present properties of b-disjunctive dominating sets in a graph. A characterization of minimal b-disjunctive dominating sets is given. We obtain bounds on the ratio of the domination number and the b-disjunctive domination number for various families of graphs, including regular graphs and trees.

Domination versus disjunctive domination in trees

A dominating set in a graph G is a set S of vertices of G such that every vertex not in S is adjacent to a vertex of S. The domination number, γ(G), of G is the minimum cardinality of a dominating set of G. A set S of vertices in G is a disjunctive dominating set in G if every vertex not in S is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it in G. The disjunctive domination number, γ2d(G), of G is the minimum cardinality of a disjunctive dominating set in G. It is known that there exist graphs G that belong to the class of bipartite graphs or claw-free graphs or chordal graphs such that γ(G)>Cγ2d(G) for any given constant C. In this paper, we show that if T is a tree, then γ(T)≤2γ2d(T)−1, and we provide a constructive characterization of the trees achieving equality in this bound.

The disjunctive domination number of a graph

For a positive integer b, we define a set S of vertices in a graph G as a b-disjunctive dominating set if every vertex not in S is adjacent to a vertex of S or has at least b vertices in S at distance 2 from it. The b-disjunctive domination number is the minimum cardinality of such a set. This concept is motivated by the concepts of distance domination and exponential domination. In this paper, we start with some simple results, then establish bounds on the parameter especially for regular graphs and claw-free graphs. We also show that determining the parameter is NP-complete, and provide a linear-time algorithm for trees.