On total isolation in graphs

Abstract

An isolating set in a graph is a set S of vertices such that removing S and its neighborhood leaves no edge; it is total isolating if S induces a subgraph with no vertex of degree 0. We show that most graphs have a partition into two disjoint total isolating sets and characterize the exceptions. Using this we show that apart from the 7-cycle, every connected graph of order \(n\ge 4\) has a total isolating set of size at most n/2, which is best possible.