Spotting Trees with Few Leaves

• We show two results related to the Hamiltonicity and k -Path algorithms in undirected graphs by Björklund [FOCS’10], and Björklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O∗(1.657^k2^{l/2}) time. It can be applied as a subroutine to solve the k -Internal Spanning Tree (k-IST) problem in O∗(min(3.455^k,1.946^n)) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time, we break the natural barrier of O∗(2^n). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree Δ=4,…,12 or with vector chromatic number at most 8.

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NATURVETENSKAP  -- Data- och informationsvetenskap -- Datavetenskap (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences -- Computer Sciences (hsv//eng)

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