This paper is motivated by the concept of nonnegative signed domination that was introduced by Huang, Li, and Feng in 2013 [15]. We study the non-negative signed domination problem from the theoretical point of view. For networks modeled by strongly chordal graphs and distance-hereditary graphs, we show that the non-negative signed domination problem can be solved in polynomial time. For networ...

Let γ(G) and ι(G) be the domination and independent domination numbers of a graph G, respectively. Introduced by Sumner and Moorer [23], a graph G is domination perfect if γ(H) = ι(H) for every induced subgraph H ⊆ G. In 1991, Zverovich and Zverovich [26] proposed a characterization of domination perfect graphs in terms of forbidden induced subgraphs. Fulman [15] noticed that this characterizat...

let $g$ be a simple graph of order $n$. the domination polynomial of $g$ is the polynomial $d(g, x)=sum_{i=gamma(g)}^{n} d(g,i) x^{i}$, where $d(g,i)$ is the number of dominating sets of $g$ of size $i$ and $gamma(g)$ is the domination number of $g$. in this paper we present some families of graphs whose domination polynomials are unimodal.

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear...

We review a characterization of domination perfect graphs in terms of forbidden induced subgraphs obtained by Zverovich and Zverovich [12] using a computer code. Then we apply it to a problem of unique domination in graphs recently proposed by Fischermann and Volkmann. 1 Domination perfect graphs Let G be a graph. A set D ⊆ V (G) is a dominating set of G if each vertex of G either belongs to D ...

In this paper we introduce the concept of edge domination and total edge domination in intuitionistic fuzzy graphs. We determine the edge domination number and total edge domination number for several classes of intuitionistic fuzzy graphs and obtain bounds for them. We also obtain Nordhaus gaddum type results for the parameters.

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we determine the domination and the total domination number in the Circulant graphs Cn〈1, 3〉, and then study γ-criticality...

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {...

For a graph G = (V,E), a set D ⊆ V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semito...