A note on the independant bondage number of planar graphs

Abstract

A vertex subset $S$ of a graph $G$ is an independent set if no two vertices in $S$ are adjacent and is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If $S$ is both independent and dominating in $G$, then $S$ is said to be an independent dominating set. The independent domination number of $G$ is the minimum cardinality among all independent dominating sets of $G$.In this paper, we investigate the independent bondage number of $G$ defined as the minimum number of edges whose removal from $G$ results in a graph with a greater independent domination number.We prove that the independent bondage number is at most 5 (respectively, 6, 7) for planar graphs with minimum degree at least 3 without cycles of lengths $4$ and $5$ (respectively, without cycles of length $4$, without intersecting triangles). Moreover, we show that every planar graph with minimum degree at least 2 and a girth at least 12 (respectively, 8 ), has an independent bondage number at most $3$ (respectively, 4).All these results improve two earlier bounds for planar graphs.