, Volume 3, Issue 1, pp 35-52

Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem

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Abstract

A(perfect) 2-matching in a graphG=(V, E) is an assignment of an integer 0, 1 or 2 to each edge of the graph in such a way that the sum over the edges incident with each node is at most (exactly) two. The incidence vector of a Hamiltonian cycle, if one exists inG, is an example of a perfect 2-matching. Fork satisfying 1≦k≦|V|, we letP k denote the problem of finding a perfect 2-matching ofG such that any cycle in the solution contains more thank edges. We call such a matching aperfect P k -matching. Then fork<l, the problemP k is a relaxation ofP 1. Moreover if |V| is odd, thenP 1V1–2 is simply the problem of determining whether or notG is Hamiltonian. A graph isP k -critical if it has no perfectP k -matching but whenever any node is deleted the resulting graph does have one. Ifk=|V|, then a graphG=(V, E) isP k -critical if and only if it ishypomatchable (the graph has an odd number of nodes and whatever node is deleted the resulting graph has a perfect matching). We prove the following results:

  1. If a graph isP k -critical, then it is alsoP l -critical for all largerl. In particular, for allk, P k -critical graphs are hypomatchable.

  2. A graphG=(V, E) has a perfectP k -matching if and only if for anyXV the number ofP k -critical components inG[V - X] is not greater than |X|.

  3. The problemP k can be solved in polynomial time provided we can recognizeP k -critical graphs in polynomial time. In addition, we describe a procedure for recognizingP k -critical graphs which is polynomial in the size of the graph and exponential ink.

Dedicated to Tibor Gallai on his seventieth birthday
Supported by NSF grant ENG 79-02506.
Supported by Sonderforschungsbereich 21 (DFG), Institut für Operations Research, Universität Bonn. and by the National Science and Engineering Research Council of Canada.