Treewidth of display graphs: bounds, brambles and applications

Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics.

[1]  Daniel H. Huson,et al.  Phylogenetic Networks - Concepts, Algorithms and Applications , 2011 .

[2]  Hans L. Bodlaender.  A linear time algorithm for finding tree-decompositions of small treewidth , 1993, STOC '93.

[3]  André Raspaud,et al.  On the vertex-arboricity of planar graphs , 2008, Eur. J. Comb..

[4]  Steven Kelk,et al.  Treewidth distance on phylogenetic trees , 2017, Theor. Comput. Sci..

[5]  Hans L. Bodlaender,et al.  A Partial k-Arboretum of Graphs with Bounded Treewidth , 1998, Theor. Comput. Sci..

[6]  T. Turner.  Phylogenetics , 2018, The International Encyclopedia of Biological Anthropology.

[7]  Mike A. Steel,et al.  Phylogeny - discrete and random processes in evolution , 2016, CBMS-NSF regional conference series in applied mathematics.

[8]  Christophe Paul,et al.  Quartets and Unrooted phylogenetic Networks , 2012, J. Bioinform. Comput. Biol..

[9]  Leo van Iersel,et al.  Phylogenetic incongruence through the lens of Monadic Second Order logic , 2015, J. Graph Algorithms Appl..

[10]  David Fernández-Baca,et al.  On compatibility and incompatibility of collections of unrooted phylogenetic trees , 2017, Discret. Appl. Math..

[11]  Frank Harary,et al.  Graph Theory , 2016 .

[12]  Leo van Iersel,et al.  Exploring the Tiers of Rooted Phylogenetic Network Space Using Tail Moves , 2018, Bulletin of mathematical biology.

[13]  Leo van Iersel,et al.  On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems , 2016, Algorithmica.

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

[15]  Steven Kelk,et al.  Constructing Minimal Phylogenetic Networks from Softwired Clusters is Fixed Parameter Tractable , 2012, Algorithmica.

[16]  M. Steel,et al.  Subtree Transfer Operations and Their Induced Metrics on Evolutionary Trees , 2001 .

[17]  Vincent Moulton,et al.  Bounds for phylogenetic network space metrics , 2017, Journal of mathematical biology.

[18]  P. Alam.  ‘L’ , 2021, Composites Engineering: An A–Z Guide.

[19]  Leo van Iersel,et al.  Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets , 2014, Algorithmica.

[20]  Bruno Courcelle,et al.  The Monadic Second-order Logic of Graphs VI: On Several Representations of Graphs by Relational Structures , 1995, Discret. Appl. Math..

[21]  Eugene M. Luks,et al.  Isomorphism of graphs of bounded valence can be tested in polynomial time , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

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

[23]  Claudia R. Solís-Lemus,et al.  Inferring Phylogenetic Networks with Maximum Pseudolikelihood under Incomplete Lineage Sorting , 2015, PLoS genetics.

[24]  Leo van Iersel,et al.  Kernelizations for the hybridization number problem on multiple nonbinary trees , 2013, J. Comput. Syst. Sci..

[25]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

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

[27]  Martin Grohe,et al.  A Faster Isomorphism Test for Graphs of Small Degree , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

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

[29]  Magnus Bordewich,et al.  On the fixed parameter tractability of agreement-based phylogenetic distances , 2017, Journal of mathematical biology.

[30]  Christophe Paul,et al.  Efficient FPT Algorithms for (Strict) Compatibility of Unrooted Phylogenetic Trees , 2016, AAIM.

[31]  Andrew Francis,et al.  Tree-Based Unrooted Phylogenetic Networks , 2018, Bulletin of mathematical biology.

[32]  Louxin Zhang,et al.  A program for verification of phylogenetic network models , 2016, Bioinform..

[33]  David Fernández-Baca,et al.  Fast Compatibility Testing for Rooted Phylogenetic Trees , 2015, Algorithmica.

[34]  M. Bordewich,et al.  Computing the Hybridization Number of Two Phylogenetic Trees Is Fixed-Parameter Tractable , 2007, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[35]  Vincent Moulton,et al.  Transforming phylogenetic networks: Moving beyond tree space. , 2016, Journal of theoretical biology.

[36]  Leo van Iersel,et al.  Locating a tree in a phylogenetic network , 2010, Inf. Process. Lett..

[37]  Arie M. C. A. Koster,et al.  Treewidth computations II. Lower bounds , 2011, Inf. Comput..

[38]  Robin Thomas,et al.  Graph Searching and a Min-Max Theorem for Tree-Width , 1993, J. Comb. Theory, Ser. B.

[39]  Gerard J. Chang,et al.  Vertex and Tree Arboricities of Graphs , 2004, J. Comb. Optim..

[40]  Sebastian Berndt.  Computing Tree Width: From Theory to Practice and Back , 2018, CiE.

[41]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[42]  Steven Kelk,et al.  On Low Treewidth Graphs and Supertrees , 2014, AlCoB.

[43]  R. Pendavingh,et al.  Reconstructing a Phylogenetic Level-1 Network from Quartets , 2013, Bulletin of Mathematical Biology.

[44]  Martin Grohe,et al.  Towards faster isomorphism tests for bounded-degree graphs , 2018, ArXiv.

[45]  Leo van Iersel,et al.  On Computing the Maximum Parsimony Score of a Phylogenetic Network , 2013, SIAM J. Discret. Math..

[46]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..