On optimal ear-decompositions of graphs

Abstract

This paper can be considered as a continuation of a paper [7] of the author. We consider optimal ear-decompositions of graphs that contain even ears as few as possible. The ear matroid of a graph was introduced in [7] via optimal ear-decompositions. Here we give a simple description of the blocks of the ear matroid of a graph. The second goal of this paper is to point out how the structural result in [7] implies easily the Tight Cut Lemma of Edmonds, Lovász and Pulleyblank. Moreover, we propose the investigation of a new class of graphs that generalizes matching-covered graphs. A graph is called ϕ-covered if each edge may lie on an even ear of an optimal ear-decomposition. Several theorems on matching-covered graphs will be generalized for ϕ-covered graphs.