On a certain class of nonideal clutters

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