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

Research output: Contribution to journalArticlepeer-review


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 Δ, we modify our algorithm to solve k-IOB in time O∗(2(2-Δ+1Δ(Δ-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.

Original languageEnglish
Pages (from-to)319-341
Number of pages23
Issue number1
StatePublished - 1 May 2017
Externally publishedYes


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

All Science Journal Classification (ASJC) codes

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics


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