Characterization of graphs of diameter 2 containing a homeomorphically irreducible spanning tree

Abstract

AbstractA spanning tree of a graph with no vertex of degree 2 is called a homeomorphically irreducible spanning tree (HIST) of the graph. In 1990, Albertson, Berman, Hutchinson, and Thomassen conjectured that every twin‐free graph with diameter 2 contains a HIST. Recently, Ando disproved this conjecture and characterized twin‐free graphs with diameter 2 that do contain a HIST. In this paper, we give a complete characterization of all graphs of diameter 2 that contain a HIST. This characterization gives alternative proofs for several known results.