Structuring the elementary components of graphs having a perfect internal matching

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Abstract

Graphs 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.

Keywords

Graph matching
Alternating path
Elementary graph
Canonical partition

Cited by (0)

1

Partially supported by Natural Science and Engineering Research Council of Canada Operating Grant 335591, and by National Scientific Research Fund of Hungary Operating Grant T 014202.

2

Author on leave from Juhász Gyula Teacher Training College, University of Szeged, Hungary. Work partially supported by the International G. Soros Foundation and by the APPOL 14084 Thematic Network project within the Fifth European Community Framework Program (FP5).