On a certain class of nonideal clutters

In this paper we define the class of near-ideal clutters following a similar concept due to Shepherd [Near perfect matrices, Math. Programming 64 (1994) 295–323] for near-perfect graphs. We prove that near-ideal clutters give a polyhedral characterization for minimally nonideal clutters as near-perfect graphs did for minimally imperfect graphs. We characterize near-ideal blockers of graphs as blockers of near-bipartite graphs. We find necessary conditions for a clutter to be near-ideal and sufficient conditions for the clutters satisfying that every minimal vertex cover is minimum.