Global Cototal Domination in Graphs

Abstract

A set of vertices D in a graph G is a dominating set, if each vertex of G is dominated by some vertices of D. The domination number $\gamma(G)$ is the minimum cardinality of a dominating set of G. A dominating set D of a graph G is a global dominating set if D is also a dominating set of $\bar{G}$. The global domination number $\gamma_g(G)$ is the minimum cardinality of a global dominating set of G. A dominating set D of a graph G is a cototal dominating set if the induced sub graph, $\langle V-D\rangle$ has no isolated vertices. The cototal domination number $\gamma_{cot}(G)$ is the minimum cardinality of a cototal dominating set of G. In this paper we introduce a new concept, the global cototal domination number $\gamma_{gcot}(G)$. A dominating set D of a graph G is a global cototal dominating set if D is both a global dominating set and a cototal dominating set. The global cototal domination number $\gamma_{gcot}(G)$ is the minimum cardinality of a global cototal domination set of G. We initiate the study of global cototal domination number and present bounds and some exact values of $\gamma_{gcot}(G)$ for some classes of graphs.