D. K. Thakkar 1 and Neha P. Jamvecha 2 *
Author Address :
1,2 Department of Mathematics, Saurashtra University, Rajkot-360005, Gujarat, India.
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.
m-dominating set, minimal m-dominating set, minimum m-dominating set, private m-neighbourhood of a vertex, m-removal of a vertex.
Article Info :
Received : October 12, 2018; Accepted : March 17, 2019.