In this paper, we generalize the theory of a convex set relaxation for the maximum weight stable set problem due to Grotschel, Lovasz and Schrijver to the generalized stable set problem. We define a convex set which serves as a relaxation problem, and show that optimizing a linear function over the set can be done in polynomial time. This implies that the generalized stable set problem for perfect bidirected graphs is polynomial time solvable. Moreover, we prove that the convex set is a polytope if and only if the corresponding bidirected graph is perfect. The definition of the convex set is based on a semidefinite programming relaxation of Lovasz and Schrijver for the maximum weight stable set problem, and the equivalent representation using infinitely many convex quadratic inequalities proposed by Fujie and Kojima is particularly important for our proof.

Bidirected graphs which are generalizations of undirected graphs, have three types of edges: (+,+)-edges, (-,-)-edges and (+,-)-edges. Undirected graphs are regarded as bidirected graphs whose edges are all of type (+,+). The notion of perfection of undirected graphs can be naturally extended to bidirected graphs in terms of polytopes. The fact that a bidirected graph is perfect if and only if the undirected graph obtained by replacing all edges to (+,+) is perfect was independently proved by several researchers. This paper gives a polyhedral proof of the fact and introduces some new knowledge on perfect bidirected graphs.

Coloring problem is a well-known combinatorial optimization problem of graphs. This paper considers H-coloring problems, which are coloring problems with restrictions such that some pairs of colors can not be used for adjacent vertices. The restriction of adjacent colors can be represented by a graph H, i.e., each vertex represents a color and each edge means that the two colors corresponding to the two end-vertices can be used for adjacent vertices. Especially, H-coloring problem with a complete graph H of order k is equivalent to the traditional k-coloring problem. This paper presents sufficient conditions such that H-coloring problem can be reduced to an H-coloring problem, where H is a subgraph of H. And it shows a hierarchy about classes of H-colorable graphs for any complement graph H of a cycle of order odd n 5.