Irredundance Trees of Diameter 3

TL;DR

This paper characterizes IR-trees of diameter 3, showing they are exactly the double stars formed by connecting the centers of two disjoint star graphs, contributing to the understanding of irredundant set structures.

Contribution

It provides a complete characterization of IR-trees with diameter 3, identifying them as double stars formed from two disjoint stars.

Findings

IR-trees of diameter 3 are precisely double stars S(2n,2n)

IR-graphs with diameter 3 are characterized as specific double star structures

The paper advances the structural understanding of irredundant set graphs

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}.