Irredundance trees of diameter 3

A set Dof vertices of a graph G =( V, E) is irredundant if each non-isolated vertex of G[ D] has a neighbour in V - Dthat 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 Ghas the IR( G)-sets as vertex set, and sets Dand D' are adjacent if and only if D' can be obtained from Dby exchanging a single vertex of Dfor an adjacent vertex in D'. An IR-tree is an IR-graph that is a tree. We characterize IR-trees of diameter 3by 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 K-1,K-2n.