Even and odd holes in cap-free graphs

It is an old problem in graph theory to test whether a graph contains a chordless cycle of length greater than three (hole) with a specific parity (even, odd). Studying the structure of graphs without odd holes has obvious implications for Berge’s strong perfect graph conjecture that states that a graph G is perfect if and only if neither G nor its complement contain an odd hole. Markossian, Gasparian, and Reed have proven that if neither G nor its complement contain an even hole, then G is β-perfect. In this article, we extend the problem of testing whether G(V, E) contains a hole of a given parity to the case where each edge of G has a label odd or even. A subset of E is odd (resp. even) if it contains an odd (resp. even) number of odd edges. Graphs for which there exists a signing (i.e., a partition of E into odd and even edges) that makes every triangle odd and every hole even are called even-signable. Graphs that can be signed so that every triangle is odd and every hole is odd are called odd-signable. We derive from a theorem due to Truemper co-NP characterizations of even-signable and oddCorrespondence to: Michele Conforti Contract grant sponsors: NSF, Laboratoire ARTEMIS, Université Joseph Fourier, Grenoble, Gruppo Nazionale Delle Ricerche-CNR Contract grant numbers: NSF DMI-9424348, DMS-9509581 and ONR N00014-89J-1063. c © 1999 John Wiley & Sons, Inc. CCC 0364-9024/99/040289-20 290 JOURNAL OF GRAPH THEORY signable graphs. A graph is strongly even-signable if it can be signed so that every cycle of length ≥ 4with at most one chord is even and every triangle is odd. Clearly a strongly even-signable graph is even-signable as well. Graphs that can be signed so that cycles of length four with one chord are even and all other cycles with at most one chord are odd are called strongly odd-signable. Every strongly oddsignable graph is odd-signable. We give co-NP characterizations for both strongly even-signable and strongly odd-signable graphs. A cap is a hole together with a node, which is adjacent to exactly two adjacent nodes on the hole. We derive a decomposition theorem for graphs that contain no cap as induced subgraph (capfree graphs). Our theorem is analogous to the decomposition theorem of Burlet and Fonlupt for Meyniel graphs, a well-studied subclass of cap-free graphs. If a graph is strongly even-signable or strongly odd-signable, then it is cap-free. In fact, strongly even-signable graphs are those cap-free graphs that are even-signable. From our decomposition theorem, we derive decomposition results for strongly odd-signable and strongly even-signable graphs. These results lead to polynomial recognition algorithms for testing whether a graph belongs to one of these classes. c © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 289–308, 1999