Graphs with isolation number equal to one third of the order

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 iota(G), is the minimum cardinality of an isolating set of G. It is known that iota(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. Published by Elsevier B.V.