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The latter ones can be partitioned, due to the fact that they contain, or do not contain, the vertices a and b, in three families, denoted by The dual solutions y * on graph G*have strictly positive components on stable sets which can be partitioned into two families denoted by Yo,"' and 9:;. A value is assigned to each of the stable sets of these families. This value is initially the (integer) component of vector y' or vector y * corresponding to the stable set. We shall now juxtapose two stable sets: one from a family defined in B,the other from a family defined in G*, in such a way as to obtain a stable set defined in G.

In other words, only one node is unsaturated. Then a graph G is hypomatchable if and only if for every v E V, there exists an np-(O,2}-matching (or a np-1-matching) which leaves u unsaturated. In [16] the following theorem was proved. 4. If G is a nonseparable hypomatchable graph, then there exist IE( np-1-matchings of G, whose incidence vectors are afinely independent. This result was proved constructively, via an algorithm which actually constructed the np-1-matchings. A shorter, nonconstructive proof of this result has been obtained by Lovhsz, which we describe here.

12). Grotschel [9] showed that those spanning edge-maximal hypohamiltonian subgraphs of K,, which satisfy a certain technical property, do induce a facet of the monotone travelling salesman polytope. 12) and the monotone polytope. 12) are facet inducing and some are not. 12) for the graph GI of Fig. 3(a) is facet inducing for Q:, but that (a) Fig. 3. (a) Facet inducing for Q:; (b) (b) Not facet inducing for 0:. The travelling salesman polytope and {O, Z}-matchings 53 Fig. 4. Modified Petersen graph.