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 K 4 − 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.
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Bonsma, P., Zickfeld, F. (2008). Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_46
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DOI: https://doi.org/10.1007/978-3-540-78773-0_46
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