Outer-connected domination in 2-connected cubic graphs

Abstract

A set S of vertices of a graph G is an outer-connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by V\S is connected. The outer-connected domination number [Formula: see text] is the minimum size of such a set. We present an infinite family of 2-connected cubic graphs, in which the number of vertices in a longest path are much less than the half of their orders. This disprove a recent conjecture posed by Akhbari, Hasni, Favaron, Karami, Sheikholeslami.