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Boyer, Geoffrey; Goddard, Wayne
Discrete Applied Mathematics, 10/2024, Letnik: 356Journal Article
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 Żyliński, 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 Lemańska, Mora, and Souto-Salorio, we determine all graphs with equality in the original bound.
