Polyhedron of triangle-free simple 2-matchings in subcubic graphs

A simple 2-matching in an edge-weighted graph is a subgraph all of whose nodes have degree 0, 1 or 2. We consider the problem of finding a maximum weight simple 2-matching that contains no triangles, which is closely related to a class of relaxations of the traveling salesman problem (TSP). Our main results are, for graphs with maximum degree 3, a complete description of the convex hull of incidence vectors of triangle-free simple 2-matchings and a strongly polynomial time algorithm for the above problem. Our system requires the use of a type of comb inequality (introduced by Grotschel and Padberg for the TSP polytope) that has {0,1,2}-coefficients and hence is more general than the well-known blossom inequality used in Edmonds’ characterization of the simple 2-matching polytope.