On m-independence in Graphs

Abstract

In this paper, we have defined the concepts of m-independent set, maximal m-independent set and maximum m-set. In order to define these concepts we have used the notion of m-adjacent vertices. Adjacent vertices are always -adjacent vertices. This notion also gives rise to a concept called m-domination in graphs. We that a set is maximal m-set if and only if it is a minimal m-dominating set. We define m-independence number of a graph to be themaximum cardinality of an m-independent set. We prove a necessary and sufficient condition under which the m-independence number decreases when a vertex is removed from the graph. Further, we have also introduced a new operation in graph called-removal of a vertex. The subgraph obtained by m-removing a vertex is a subgraph of the subgraph obtained by removing the vertex from the graph. We prove that a vertex is an isolated vertex if and only if the m-independence number of the graphdecreases when the vertex is m-removed from the graph. Some related examples have been given to illustrate these concepts.