: This paper shows some useful properties of the adjacency structures of a class of combinatorial polyhedra including the equality constrained 0-1 polytopes. The class of polyhedra considered here includes 0-1 polytopes related to some combinatorial optimization problems; e.g., set partitioning polytopes, set packing polytopes, perfect matching polytopes, vertex packing polytopes and all the faces of these polytopes. First, we establish two fundamental properties of the equality constrained 0-1 polytopes. This paper deals with the polyhedra satisfying these two fundamental properties. We consider a path on the polyhedron satisfying the condition that for each co-ordinate, the vertices in a path form a monotonic sequence. When one of the end vertices of the path is optimal to an optimization problem defined on the polyhedron, the associated objective values form a monotonic sequence and the length of the path is bounded by the dimension of the polytope. In a sense, some of the results i...