On Low Treewidth Graphs and Supertrees

Compatibility of unrooted phylogenetic trees is a well studied problem in phylogenetics. It asks to determine whether for a set of k input trees T 1,...,T k there exists a larger tree (called a supertree) that contains the topologies of all k input trees. When any such supertree exists we call the instance compatible and otherwise incompatible. It is known that the problem is NP-hard and FPT, although a constructive FPT algorithm is not known. It has been shown that whenever the treewidth of an auxiliary structure known as the display graph is strictly larger than the number of input trees, the instance is incompatible. Here we show that whenever the treewidth of the display graph is at most 2, the instance is compatible. Furthermore, we give a polynomial-time algorithm to construct a supertree in this case. Finally, we demonstrate both compatible and incompatible instances that have display graphs with treewidth 3, highlighting that the treewidth of the display graph is (on its own) not sufficient to determine compatibility.

[1]  Derek G. Corneil,et al.  Complexity of finding embeddings in a k -tree , 1987 .

[2]  O. Bininda-Emonds.  Phylogenetic Supertrees: Combining Information To Reveal The Tree Of Life , 2004 .

[3]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[4]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[5]  David Fernández-Baca,et al.  Characterizing compatibility and agreement of unrooted trees via cuts in graphs , 2013, Algorithms for Molecular Biology.

[6]  M. Steel.  The complexity of reconstructing trees from qualitative characters and subtrees , 1992 .

[7]  Tandy J. Warnow,et al.  Approximating the Complement of the Maximum Compatible Subset of Leaves of k Trees , 2002, APPROX.

[8]  Hans L. Bodlaender,et al.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC.

[9]  Katharina T. Huber,et al.  Basic Phylogenetic Combinatorics , 2011 .

[10]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs X: Linear Orderings , 1996, Theor. Comput. Sci..

[11]  David Bryant,et al.  Compatibility of unrooted phylogenetic trees is FPT , 2006, Theor. Comput. Sci..

[12]  Dan Gusfield,et al.  Reducing Problems in Unrooted Tree Compatibility to Restricted Triangulations of Intersection Graphs , 2012, WABI.

[13]  S. Griffis.  EDITOR , 1997, Journal of Navigation.

[14]  David Fernández-Baca,et al.  Graph triangulations and the compatibility of unrooted phylogenetic trees , 2010, Appl. Math. Lett..

[15]  Alfred V. Aho,et al.  Inferring a Tree from Lowest Common Ancestors with an Application to the Optimization of Relational Expressions , 1981, SIAM J. Comput..

[16]  Arie M. C. A. Koster,et al.  Treewidth computations I. Upper bounds , 2010, Inf. Comput..

[17]  Charles Semple,et al.  Quartet Compatibility and the Quartet Graph , 2008, Electron. J. Comb..

[18]  R. Ho.  Algebraic Topology , 2022 .