Irredundance trees of diameter 3

Abstract

A set D of vertices of a graph G=(V,E) is irredundant if each non-isolated vertex of G[D] has a neighbour in V−D that is not adjacent to any other vertex in D. The upper irredundance number IR(G) is the largest cardinality of an irredundant set of G; an IR(G)-set is an irredundant set of cardinality IR(G).The IR-graph of G has the IR(G)-sets as vertex set, and sets D and D′ are adjacent if and only if D′ can be obtained from D by exchanging a single vertex of D for an adjacent vertex in D′. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3 by showing that these graphs are precisely the double stars S(2n,2n), i.e., trees obtained by joining the central vertices of two disjoint stars K1,2n.