Exact double domination in graphs (original) (raw)

Exact Double Domination in Graph

International Journal of computing Algorithm, 2014

In this paper, we deals about exact double domination in graphs. In a graph a vertex is said to dominate itself and all its neighbours. A double dominating set is exact if every vertex of G is dominated exactly twice. If a double dominating set exist then all such sets have the same size and bounds on this size. We established a necessary and sufficient condition of exact double dominating set in a connected cubic graph with application.

On Two Open Problems on Double Vertex-Edge Domination in Graphs

Mathematics

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those g...

Generalized perfect domination in graphs

Journal of Combinatorial Optimization, 2014

Let k be a positive integer and G = (V , E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ kp (G). In this paper, we give characterizations of graphs for which γ kp (G) = γ (G) + k − 2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.

Algorithmic Aspects of Some Variants of Domination in Graphs

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2020

A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the indep...

Perfect dominating sets

A dominating set Ë of a graph is perfect if each vertex of is dominated by exactly one vertex in Ë. We study the existence and construction of PDSs in families of graphs arising from the interconnection networks of parallel computers. These include trees, dags, series-parallel graphs, meshes, tori, hypercubes, cube-connected cycles, cube-connected paths, and de Bruijn graphs. For trees, dags, and series-parallel graphs we give linear time algorithms that determine if a PDS exists, and generate a PDS when one does. For 2-and 3-dimensional meshes, 2-dimensional tori, hypercubes, and cube-connected paths we completely characterize which graphs have a PDS, and the structure of all PDSs. For higher dimensional meshes and tori, cube-connected cycles, and de Bruijn graphs, we show the existence of a PDS in infinitely many cases, but our characterization is not complete. Our results include distance -domination for arbitrary .

Degree equitable restrained double domination in graphs

2021

A subset D ⊆ V ( G ) is called an equitable dominating set of a graph G if every vertex v ∈ V ( G ) \ D has a neighbor u ∈ D such that | d G ( u )- d G ( v )| ≤ 1. An equitable dominating set D is a degree equitable restrained double dominating set (DERD-dominating set) of G if every vertex of G is dominated by at least two vertices of D , and 〈 V ( G ) \ D 〉 has no isolated vertices. The DERD-domination number of G , denoted by γ cl ^ e ( G ), is the minimum cardinality of a DERD-dominating set of G . We initiate the study of DERD-domination in graphs and we obtain some sharp bounds. Finally, we show that the decision problem for determining γ cl ^ e ( G ) is NP-complete.

Notions of Domination for Some Classes of Graphs

2016

IfG = (V,E) is a finite simple connected graph, a subset S of V is said to be a dominating set of the graph G if every vertex of G is either in S or is adjacent to at least one element of S. The minimum cardinality of a dominating set of G is called the domination number of G. In the present study, we consider variations of the concept of domination in a graph. A subset S of V is said to be a triple connected dominating set of G if S is a dominating set and any set of three vertices in the subgraph 〈S〉 induced by S lie in a common path. In this case, we say that the graph is triple connected. The minimum cardinality of a triple connected dominating set is called the triple connected domination number of G and is denoted by γtc(G). On the other hand, a subset S of V of a nontrivial connected graph G is said to be a triple connected complementary tree dominating set, if S is a triple connected dominating set and the induced subgraph 〈V − S〉 is a tree. The minimum cardinality of a trip...

Super Dominating Sets in Graphs

Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2015

Let G = (V, E) be a graph. A subset D of V (G) is called a super dominating set if for every v ∈ V (G) − D there exists an external private neighbour of v with respect to V (G) − D. The minimum cardinality of a super dominating set is called the super domination number of G and is denoted by γsp(G). In this paper some results on the super domination number are obtained. We prove that if T is a tree with at least three vertices, then n 2 ≤ γsp(T) ≤ n − s, where s is the number of support vertices in T and we characterize the extremal trees.

Some results on the exact 1-step domination graphs

Mathematica Montisnigri

An exact 1-step dominating set in a graph G is a subset S of vertices of G such that () 1 N v S   for every vertex () v V G . A graph is an exact 1-step domination graph if it contains an exact 1-step dominating set. In this paper, we obtain new upper bounds on the size of exact 1-step domination graphs. We also present an upper bound on the total domination number of an exact 1-step domination tree and characterize trees achieving equality for this bound.

On -total domination in graphs

Discrete Applied Mathematics, 2012

Let G = (V , E) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S. For some α with 0 < α ≤ 1, a total dominating set S in G is an α-total dominating set if for every vertex v ∈ V \ S, |N(v) ∩ S| ≥ α|N(v)|. The minimum cardinality of an α-total dominating set of G is called the α-total domination number of G. In this paper, we study α-total domination in graphs. We obtain several results and bounds for the α-total domination number of a graph G.