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, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

Let G = (V, E) be a graph. Let D be a minimum secure total dominating set of G. If V – D contains a secure total dominating set D' of G, then D' is called an inverse secure total dominating set with respect to D. The inverse secure total domination number γst(G) of G is the minimum cardinality of an inverse secure total dominating set of G. The disjoint secure total domination number γstγst(G) ...

We show that the total domination number of a graph G whose complement , G c , does not contain K3;3 is at most (G c), except for complements of complete graphs, and graphs belonging to a certain family which is characterized. In the case where G c does not contain K4;4 we show that there are four exceptional families of graphs, and determine the total domination number of the graphs in each one.

We limit our discussion to graphs that are simple and finite of order . Although 8 we often identify a graph with its set of vertices, in cases where we need to be K explicit we write . A set of vertices of is said to Z ÐKÑ Q K dominate K provided each vertex of is either in or adjacent to a vertex of . K Q Q The domination number of is the minimum order of a dominating set. A K dominating prov...

A edge 2-rainbow dominating function (E2RDF) of a graph G is a ‎function f from the edge set E(G) to the set of all subsets‎ ‎of the set {1,2} such that for any edge.......................

A set S V is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. A set S V is an independent set of vertices if no two vertices in S are adjacent. The independence number, B0 (G), is the maximum cardinalit...

‎the annihilator graph $ag(r)$ of a commutative ring $r$ is a simple undirected graph with the vertex set $z(r)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎in this paper we give the sufficient condition for a graph $ag(r)$ to be complete‎. ‎we characterize rings for which $ag(r)$ is a regular graph‎, ‎we show that $gamma (ag(r))in {1,2}$ and...

Given a graph \(G=(V(G), E(G))\), the size of minimum dominating set, paired and total set G are denoted by \(\gamma (G)\), _{pr}(G)\), _{t}(G)\), respectively. For positive integer k, k-packing in is \(S \subseteq V(G)\) such that for every pair distinct vertices u v S, distance between at least \(k+1\). The number order largest \(\rho _{k}(G)\). It well known _{pr}(G) \le 2\gamma (G)\). In th...