Splitters and barriers in open graphs having a perfect internal matching

A counterpart of Tutte's Theorem and Berge's formula is proved for open graphs with perfect (maximum) internal matchings. Properties of barriers and factor-critical graphs are studied in the new context, and an efficient algorithm is given to find maximal barriers of graphs having a perfect internal matching

Structuring the elementary components of graphs having a perfect internal matching

AbstractGraphs with perfect internal matchings are decomposed into elementary components, and these components are given a structure reflecting the order in which they can be reached by external alternating paths. It is shown that the set of elementary components can be grouped into pairwise disjoint families determined by the “two-way accessible” relationship among them. A family tree is established by which every family member, except the root, has a unique father and mother identified as another elementary component and one of its canonical classes, from which the given member is two-way accessible. It is proved that every member of the family is only accessible through a distinguished canonical class of the root by external alternating paths. The families themselves are arranged in a partial order according to the order they can be covered by external alternating paths, and a complete characterization of the graph's forbidden and impervious edges is elaborated