On the spanning trees of weighted graphs


Given a weighted graph, letW 1,W 2,W 3,... denote the increasing sequence of all possible distinct spanning tree weights. Settling a conjecture due to Kano, we prove that every spanning tree of weightW 1 is at mostk−1 edge swaps away from some spanning tree of weightW k . Three other conjectures posed by Kano are proven for two special classes of graphs. Finally, we consider the algorithmic complexity of generating a spanning tree of weightW k .

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  1. [1]

    J. Edmonds: Systems of distinct representatives and linear algebra,J. of Research and the National Bureau of Standards 71B (1967), 241–245.

    Google Scholar 

  2. [2]

    M. L. Fredman, andR. E. Tarjan: Fibonacci heaps and their uses in improved network optimization algorithms,JACM 34 (1987), 596–615.

    Google Scholar 

  3. [3]

    D. B. Johnson, andS. D. Kashdan: Lower bounds for selection inX+Y and other multisets,JACM 25 (1978), 556–570.

    Google Scholar 

  4. [4]

    Y. Kajitani: Graph theoretical properties of the node determinant of an LCR network,IEEE Trans. Circuit Theory CT-18 (1971), 343–350.

    Google Scholar 

  5. [5]

    M. Kano: Maximum andkth maximal spanning trees of a weighted graph,Combinatorica 7 (1987), 205–214.

    Google Scholar 

  6. [6]

    T. Kawamoto, Y. Kajitani, andS. Shinoda: On the second maximal spanning trees of a weighted graph (in Japanese),Trans. IECE of Japan 61A (1978), 988–995.

    Google Scholar 

  7. [7]

    D. E. Knuth:The Art of Computer Programming Vol. I: Fundamental Algorithms, Addison-Wesley, Reading, Mass.

  8. [8]

    J. B. Kruskal: On the shortest spanning subtree of a graph and the traveling salesman problem,Proc. Amer. Math. Soc. 7 (1956), 48–50.

    Google Scholar 

  9. [9]

    E. L. Lawler: A procedure for computing theK best solutions to discrete optimization problems and its application to the shortest path problem,Management Sci. 18 (1972), 401–405.

    Google Scholar 

  10. [10]

    Okada, andOnodera:Bull. Yamagata Univ. 2 (1952), 89–117 (cited in [7]).

    Google Scholar 

  11. [11]

    R. C. Prim: Shortest connection networks and some generalizationsBell System Technical J. 36 (1957), 1389–1401.

    Google Scholar 

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Additional information

This work was supported in part by a grant from the AT&T foundation and NSF grant DCR-8351757.

Primarily supported by a 1967 Science and Engineering Scholarship from the Natural Sciences and Engineering Research Council of Canada.

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Mayr, E.W., Plaxton, C.G. On the spanning trees of weighted graphs. Combinatorica 12, 433–447 (1992). https://doi.org/10.1007/BF01305236

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AMS subject classification code (1991)

  • 05 C 05
  • 05 C 85
  • 68 R 10