Complementary Tree Nil Domination Number of Splitting Graphs

Abstract

A set D of a graph $G = (V, E)$ is a dominating set, if every vertex in $V(G)-D$ is adjacent to some vertex in D. The domination number $\gamma(G)$ of G is the minimum cardinality of a dominating set. A dominating set D is called a complementary tree nil dominating set, if $V(G)-D$ is not a dominating set and also the induced subgraph $\langle V(G)-D\rangle$ is a tree. The minimum cardinality of a complementary tree nil dominating set is called the complementary tree nil domination number of G and is denoted by $\gamma_{ctnd}(G)$. In this paper, some results regarding the complementary tree nil domination number of splitting graphs of connected graphs are found.