Irredundance graphs

Abstract

A set D of vertices of a graph G = ( V , E ) is irredundant if each v ∈ 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 , n , n ≥ 2 , P 4 , P 5 , C 5 , C 6 , C 7 ) 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.