Automatic Control and Computer Sciences

, Volume 51, Issue 7, pp 682–688 | Cite as

1-Skeletons of the Spanning Tree Problems with Additional Constraints

  • V. A. Bondarenko
  • A. V. Nikolaev
  • D. A. Shovgenov


We consider the 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 or equal a given value. In the second problem, an additional constraint is the assumption that the degree of all vertices of the spanning tree does not exceed a given value. The decision versions of both problems are NP-complete. We consider the 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 in combinatorial and geometric properties between the considered problems and the classical minimum spanning tree problem.


spanning tree 1-skeleton clique number NP-complete problem Hamiltonian path 


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Copyright information

© Allerton Press, Inc. 2017

Authors and Affiliations

  • V. A. Bondarenko
    • 1
  • A. V. Nikolaev
    • 1
  • D. A. Shovgenov
    • 1
  1. 1.Yaroslavl State UniversityYaroslavlRussia

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