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arXiv · 2401.03933

Disjoint Isolating Sets and Graphs with Maximum Isolation Number

Geoffrey Boyer·Wayne Goddard
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Abstract

An isolating set in a graph is a set $X$ of vertices such that every edge of the graph is incident with a vertex of $X$ or its neighborhood. The isolation number of a graph, or equivalently the vertex-edge domination number, is the minimum number of vertices in an isolating set. Caro and Hansberg, and independently \.{Z}yli\'{n}ski, showed that the isolation number is at most one-third the order for every connected graph of order at least $6$. We show that in fact all such graphs have three disjoint isolating sets. Further, using a family introduced by Lema\'{n}ska, Mora, and Souto-Salorio, we determine all graphs with equality in the original bound.

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Geoffrey Boyer, Wayne Goddard. 2024-01-08. Disjoint Isolating Sets and Graphs with Maximum Isolation Number. https://arxiv.org/abs/2401.03933

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