In this paper, we have defined the concepts of m-independent set, maximal m-independent set and maximum mindependent set. In order to define these concepts we have used the notion of m-adjacent vertices. Adjacent vertices are always m-adjacent vertices. This notion also gives rise to a concept called m-domination in graphs. We prove that a set is maximal mindependent set if and only if it is a minimal m-dominating set. We define m-independence number of a graph to be the maximum 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 m-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 graph decreases when the vertex is m-removed from the graph. Some related examples have been given to illustrate these concepts.

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Abstract. The paper is about the ve-domination (vertex-edge domination) in graphs. Necessary and sufficient conditions are proved under which the ve-domination number decreases or increases.

In this paper, we have defined the concept of m-dominating set in graphs. In order to define this concept we have used the notion of m-adjacent vertices. We have also defined the concepts of minimal m-dominating set, minimum m-dominating set and m-domination number which is the minimum cardinality of an m-dominating set. We prove that the complement of a minimal m-dominating set is an m-dominating set. Also we prove a necessary and sufficient condition under which the m-domination number increases or decreases when a vertex is removed from the graph. Further we have also studied the concept of m-removing a vertex from the graph and we prove that the m-removal of a vertex from the graph always increases or does not change the m-domination number. Some examples have also been given. Keywords m-dominating set, minimal m-dominating set, minimum m-dominating set, private m-neighbourhood of a vertex, m-removal of a vertex.

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