Graphs with isolation number equal to one third of the order

Abstract

[Absctract]: A set D of vertices of a graph G is isolating if the set of vertices not in D and with no neighbor in D is independent. The isolation number of G, denoted by ι(G), is the minimum cardinality of an isolating set of G. It is known that ι(G) ≤ n/3, if G is a connected graph of order n, n ≥ 3, distinct from C5. The main result of this work is the characterisation of unicyclic and block graphs of order n with isolating number equal to n/3. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.