Independent and Vertex Covering Number on Modular Product of Simple Graph

Abstract

The Modular Product 1 2 G G of graph of 1 G and 2 G has vertex set 1 2 1 2 V(G G )=V(G )×V(G ) and edge set    1 2 1 1 2 2 1 2 1 1 2 2 E(G G )  (u ,v )(u ,v ) u u E(G ) and v v E(G )  1 2 1 1 2 u u E(G ) and v v  2 E(G ) . A subset U of the vertex set VG of G is said to be an independent set of G if the induced subgraph GU is a trivial graph. An independent set of G with maximum number of vertices is called a maximum independent set of G . The number of vertices of a maximum independent set of G is called the independent number of G . A vertex of graph G is said to cover the edges incident with it, and a vertex cover of a graph G is a set of vertices covering all the edges of G . The minimum cardinality of a vertex cover of a graph G is called the vertex covering number of G . A dominating set (or domset) of graph G is a subset D of VG such that each vertex of V  D is adjacent to at least one vertex of D . The minimum cardinality of a dominating set of a graph G is called the domination number of G . In this research, we determine generalizations of some of the vertex-graph parameters are independent number, vertex covering number and domination number on Modular product of simple graph and complete bipartite graph.