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arXiv · 2104.09004

Irredundance Trees of Diameter 3

C. M. Mynhardt·A. Roux
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Abstract

A set D of vertices of a graph G with vertex set V 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 A and B are adjacent if and only if B can be obtained from A by exchanging a single vertex of A for an adjacent vertex in B. 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 K_{1,2n}.

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C. M. Mynhardt, A. Roux. 2021-04-19. Irredundance Trees of Diameter 3. https://arxiv.org/abs/2104.09004

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