Independent bondage number of a graph

Abstract

A vertex set S of a simple finite graph $G=(V;E)$ is said to be an independent set if there is no edge between any pair of vertices of S and a dominating set if for any $v\in V-S$, $uv\in E$ for some $u\in S$. If S is both independent and dominating in G, then S is an independent dominating set. Let i(G) denote the cardinality of a minimum independent dominating set of G. Set $b_i(G)=\min \{|E'|~: E'\subseteq E, i(G)<i(G-E')\}$, which we define as an independent bondage number of G. We initiate the study on properties of $b_i(G)$ and present some upper and lower bounds on $b_i(G)$ for some special classes of graphs. Further research problems will be proposed.