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Algorithms for k-Internal Out-Branching and k-Tree in Bounded Degree Graphs

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In this paper, we employ the multilinear detection technique, combined with proper colorings of graphs, to develop algorithms for two problems in bounded degree graphs. We focus mostly on the k-Internal Out-Branching (k-IOB) problem, which asks if a given directed graph has an out-branching (i.e., a spanning tree with exactly one node of in-degree 0) with at least k internal nodes. The second problem, k-Tree, asks if a given undirected graph G has a (not necessarily induced) copy of a given tree T. That is, k-Tree asks whether T is a subgraph of G. We present an \(O^*(4^k)\) time randomized algorithm for k-IOB, which improves the \(O^*\) running time of the previous best known algorithm for this problem. Then, for directed graphs whose underlying (simple, undirected) graphs have bounded degree \(\varDelta \), we modify our algorithm to solve k-IOB in time \(O^*(2^{(2-\frac{\varDelta +1}{\varDelta (\varDelta -1)})k})\). For k- Tree in graphs of bounded degree 3, we obtain an \(O^*(1.914^k)\) time randomized algorithm. In particular, all of our algorithms use polynomial space.

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Fig. 1


  1. 1.

    Formal definitions of concepts mentioned in the introduction are given in Sect. 2.

  2. 2.

    \(O^*\) hides factors polynomial in the input size.

  3. 3.

    Rcent developments related to k-IOB are discussed in Sect. 6.


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Correspondence to Meirav Zehavi.

Additional information

A preliminary version of this paper appeared in the proceedings of the 8th International Symposium on Parameterized and Exact Computation (IPEC’13) [26].

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Zehavi, M. Algorithms for k-Internal Out-Branching and k-Tree in Bounded Degree Graphs. Algorithmica 78, 319–341 (2017). https://doi.org/10.1007/s00453-016-0166-3

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  • Parameterized algorithm
  • k-Internal out-branching
  • k-Tree
  • Multilinear detection
  • Proper coloring