1-Skeletons of the spanning tree problems with additional constraints

In this paper, we study polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less than or equal to a given value. In the second problem, an additional constraint is the assumption that the degree of all nodes of the spanning tree does not exceed a given value. The recognition versions of both problems are NP-complete. We consider polytopes of these problems and their 1-skeletons. We prove that in both cases it is a NP-complete problem to determine whether the vertices of 1-skeleton are adjacent. Although it is possible to obtain a superpolynomial lower bounds on the clique numbers of these graphs. These values characterize the time complexity in a broad class of algorithms based on linear comparisons. The results indicate a fundamental difference between combinatorial and geometric properties of the considered problems from the classical minimum spanning tree problem.