Reduced graphs of diameter two

Abstract

AbstractA graph H is collapsible if for every subset X ⊆ V(H), H has a spanning connected subgraph whose set of odd‐degree vertices is X. In any graph G there is a unique collection of maximal collapsible subgraphs, and when all of them are contracted, the resulting contraction of G is a reduced graph. Interest in reduced graphs arises from the fact [4] that a graph G has a spanning closed trail if and only if its corresponding reduced graph has a spanning closed trail. The concept can also be applied to study hamiltonian line graphs [11] or double cycle covers [8]. In this article, we characterize the reduced graphs of diameter two. As applications, we obtain prior results in [12] and [14], and show that every 2‐edge‐connected graph with diameter at most two either admits a double cycle cover with three even subgraphs or is isomorphic to the Petersen graph.