Spanning trees with many leaves in graphs without diamonds and blossoms

  • Authors:

  • Paul Bonsma;Florian Zickfeld

  • Affiliations:

  • Technische Universität Berlin, Berlin, Germany;Technische Universität Berlin, Berlin, Germany

  • Venue:
  • LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
  • Year:
  • 2008

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Abstract

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K4 - e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n + 4)/3 leaves. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3 + c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O*(6.75k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for Max-Leaves Spanning Tree.