ON SOME RESULTS OF PERFECT DOMINATIONS OF SOME GRAPHS

ON SOME RESULTS OF PERFECT DOMINATIONS OF SOME GRAPHS

M. L. Caay, S. R. Palahang

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Abstract

A dominating set D   V (G) of a simple graph G is the set of all u such that for every v 2 V (G)nD, uv 2 E(G). An independent set I   V (G) is a set of non-adjacent vertices in G. An independent dominating set Di   V (G) is a subset of V (G) that is both independent set and dominating set. A subset S of V (G) is called a perfect dominating set of S if for each v belongs to V (G)nS, there exists a unique element u 2 S, such that v and u are adjacent. De ne an independent perfect dominating set Dip of G to be a dominating set that is both independent dominating set and perfect dominating set. The minimum cardinality of an independent perfect dominating set of G is called an independent perfect domination number of G, denoted by ip(G). If a graph has a perfect dominating set, we say that the graph G is ip-graph. In this study, we determine some bounds and parameters of the graph as well as the existence existence of this invariant to some graphs and graphs formed by some binary operations.

Keywords

perfect dominating set, independent dominating set, independent perfect dominating set, independent perfect domination number.