Irredundance graphs

A set D of vertices of a graph G = (V, E) is irredundant if each v is an element of D satisfies (a) v is isolated in the subgraph induced by D, or (b) v is adjacent to a vertex in V - D that is nonadjacent to all other vertices 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 irredundant sets of G of maximum cardinality, that is, the IR(G)-sets, as vertex set, and sets D and D ' are adjacent if and only if D ' is obtained from D by exchanging a single vertex of D for an adjacent vertex in D '. We study the realizability of graphs as IR-graphs and show that all disconnected graphs are IR-graphs, but some connected graphs (e.g. stars K-1,K-n, n >= 2, P-4, P-5, C-5, C-6, C7) are not. We show that the double star S(2, 2) - the tree obtained by joining the two central vertices of two disjoint copies of P-3 - is the unique smallest IR-tree with diameter 3 and also a smallest non-complete IR-tree, and the tree obtained by subdividing a single pendant edge of S(2, 2) is the unique smallest IR-tree with diameter 4. (C) 2022 Elsevier B.V. All rights reserved.