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.......................

‎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...

The open neighborhood of a vertex $v$ of a graph $G$ is the set $N(v)$ consisting of all vertices adjacent to $v$ in $G$. For $Dsubseteq V(G)$, we define $overline{D}=V(G)setminus D$. A set $Dsubseteq V(G)$ is called a super dominating set of $G$ if for every vertex $uin overline{D}$, there exists $vin D$ such that $N(v)cap overline{D}={u}$. The super domination number of $G$ is the minimum car...

‎A Roman dominating function (RDF) on a graph G=(V,E) is a function  f : V → {0, 1, 2}  such that every vertex u for which f(u)=0 is‎ ‎adjacent to at least one vertex v for which f(v)=2‎. ‎An RDF f is called‎‎an outer independent Roman dominating function (OIRDF) if the set of‎‎vertices assigned a 0 under f is an independent set‎. ‎The weight of an‎‎OIRDF is the sum of its function values over ...

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D. The geodetic number of G is the minimum cardinality of a geodetic set of G. We prove that it is NP complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k. Furthermore...

For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the ...

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products hav...

-Given two vertices u and v of a connected graph G=(V, E), the closed interval I[u, v] is that set of all vertices lying in some u-v geodesic in G. A subset of V(G) S={v1,v2,v3,....,vk} is a linear geodetic set or sequential geodetic set if each vertex x of G lies on a vi – vi+1 geodesic where 1 ≤ i < k . A linear geodetic set of minimum cardinality in G is called as linear geodetic number lgn(...

For a nontrivial connected graph G = (V (G), E(G)), a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a speci...