Minimally Non-Preperfect Graphs of Small Maximum Degree

A graph G is called preperfect if each induced subgraph G 0 G of order at least 2 has two vertices x; y such that either all maximum cliques of G 0 containing x contain y, or all maximum independent sets of G 0 containing y contain x, too. Giving a partial answer to a problem of Hammer and Maaray Combinatorica 13 (1993), 199{208], we describe new classes of minimally non-preperfect graphs, and prove the following characterizations : (i) A graph of maximum degree 4 is minimally non-preperfect if and only if it is an odd cycle of length at least 5, or the complement of a cycle of length 7, or the line graph of a 3-regular 3-connected bipartite graph. (ii) If a graph G is not an odd cycle and has no isolated vertices, then its line graph is minimally non-preperfect if and only if G is bipartite, 3-edge-connected, regular of degree d for some d 3, and contains no 3-edge-connected d 0-regular subgraph for any 3 d 0 < d.