ON SLIM GRAPHS, EVEN PAIRS, AND STAR-CUTSETS

Meyniel proved that a graph G is perfect if every odd cycle of G with at least five vertices has at least two chords. A slim graph is any graph obtained from a Meyniel graph by removing all the edges of a given induced subgraph. Hertz introduced slim graphs and proved that they are perfect. We show that Hertz's result can be derived from a deep characterization of Meyniel graphs which is due to Burlet and Fonlupt. Hertz also asked whether every slim graph which is not a clique has an even pair of vertices, and whether every nonbipartite slim graph has a star-cutset. We provide partial solutions to these questions for slim graphs that are derived from i-triangulated graphs and parity graphs.