Abstract
Tree representations of (sets of) symmetric binary relations, or equivalently edge-colored undirected graphs, are of central interest, e.g. in phylogenomics. In this context symbolic ultrametrics play a crucial role. Symbolic ultrametrics define an edge-colored complete graph that allows to represent the topology of this graph as a vertex-colored tree. Here, we are interested in the structure and the complexity of certain combinatorial problems resulting from considerations based on symbolic ultrametrics, and on algorithms to solve them.This includes, the characterization of symbolic ultrametrics that additionally distinguishes between edges and non-edges of arbitrary edge-colored graphs G and thus, yielding a tree representation of G, by means of so-called cographs. Moreover, we address the problem of finding “closest” symbolic ultrametrics and show the NP-completeness of the three problems: symbolic ultrametric editing, completion and deletion. Finally, as not all graphs are cographs, and hence, do not have a tree representation, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of an arbitrary non-cograph G. This is equivalent to find an optimal cograph edge k-decomposition \(\{E_1,\dots ,E_k\}\) of E so that each subgraph \((V,E_i)\) of G is a cograph. We investigate this problem in full detail, resulting in several new open problems, and NP-hardness results.For all optimization problems proven to be NP-hard we will provide integer linear program formulations to solve them.
Similar content being viewed by others
References
Achlioptas D (1997) The complexity of g-free colourability. Discrete Math 165166:21–30 (Graphs and combinatorics)
Bernini A, Ferrari L, Pinzani R (2009) Enumeration of some classes of words avoiding two generalized patterns of length three. J Autom Lang Comb 14(2):129–147
Bilotta S, Grazzini E, Pergola E (2013) Counting binary words avoiding alternating patterns. J Integer Seq 16:Article 13.4.8
Böcker S, Dress A (1998) Recovering symbolically dated, rooted trees from symbolic ultrametrics. Adv Math 138:105–125
Brändén P, Mansour T (2005) Finite automata and pattern avoidance in words. J Comb Theory Ser A 110(1):127–145
Brandstädt A, Le V, Spinrad J (1999) Graph classes: a survey SIAM monographs on discrete mathematics and applications. Society of Industrial and Applied Mathematics, Philadephia
Burstein A, Mansour T (2002) Words restricted by patterns with at most 2 distinct letters. Electron J Comb Number Theory 9(2):1–16
Corneil D, Perl Y, Stewart L (1985) A linear recognition algorithm for cographs. SIAM J Comput 14(4):926–934
Corneil DG, Lerchs H, Stewart Burlingham LK (1981) Complement reducible graphs. Discrete Appl Math 3:163–174
Dorbec P, Montassier M, Ochem P (2014) Vertex partitions of graphs into cographs and stars. J Graph Theory 75(1):75–90
El-Mallah E, Colbourn C (1988) The complexity of some edge deletion problems. IEEE Trans Circuits Syst 35(3):354–362
Fitch W (2000) Homology: a personal view on some of the problems. Trends Genet 16:227–231
Gimbel J, Nesětrǐl J (2002) Partitions of graphs into cographs. Electronic notes in discrete mathematics, vol 11. In: The ninth quadrennial international conference on graph theory, combinatorics, algorithms and applications, pp 705–721
Hammack R, Imrich W, Klavžar S (2011) Handbook of product graphs, 2nd edn. CRC Press, Boca Raton
Hellmuth M, Hernandez-Rosales M, Huber KT, Moulton V, Stadler PF, Wieseke N (2013a) Orthology relations, symbolic ultrametrics, and cographs. J Math Biol 66(1–2):399–420
Hellmuth M, Imrich W, Kupka T (2013b) Partial star products: a local covering approach for the recognition of approximate Cartesian product graphs. Math Comput Sci 7(3):255–273
Hellmuth M, Wieseke N (2015) On symbolic ultrametrics, cotree representations, and cograph edge decompositions and partitions. In: Xu D, Du D, Du D (eds) Computing and combinatorics. Lecture Notes in Computer Science, vol 9198. Springer International Publishing, Cham, pp 609–623
Hellmuth M, Wieseke N (2016) From sequence data Including orthologs, paralogs, and xenologs to gene and species trees, Chap. 21. Springer International Publishing, Cham
Hellmuth M, Wieseke N, Lechner M, Lenhof H, Middendorf M, Stadler P (2015) Phylogenomics with paralogs. Proc Natl Acad Sci 112(7):2058–2063
Hernandez-Rosales M, Hellmuth M, Wieseke N, Huber KT, Moulton V, Stadler PF (2012) From event-labeled gene trees to species trees. BMC Bioinform 13(19):S6
Lafond M, El-Mabrouk N (2014) Orthology and paralogy constraints: satisfiability and consistency. BMC Genomics 15(Suppl 6):S12. doi:10.1186/1471-2164-15-S6-S12.
Lechner M, Findeiß S, Steiner L, Marz M, Stadler PF, Prohaska SJ (2011) Proteinortho: detection of (co-)orthologs in large-scale analysis. BMC Bioinform 12:124
Lechner M, Hernandez-Rosales M, Doerr D, Wiesecke N, Thevenin A, Stoye J, Hartmann RK, Prohaska SJ, Stadler PF (2014) Orthology detection combining clustering and synteny for very large datasets. PLoS ONE 9(8):e105,015
Lerchs H (1971a) On cliques and kernels. Technical Report, Department of Computer Science University of Toronto
Lerchs H (1971b) On the clique–kernel structure of graphs. Technical Report, Department of Computer Science, University of Toronto
Liu Y, Wang J, Guo J, Chen J (2011) Cograph editing: complexity and parametrized algorithms. In: Fu B, Du DZ (eds) COCOON 2011, Lecture Notes in Computer Science, vol 6842. Springer, Berlin, Heidelberg, pp 110–121
Liu Y, Wang J, Guo J, Chen J (2012) Complexity and parameterized algorithms for cograph editing. Theoret Comput Sci 461:45–54
Misra J, Gries D (1992) A constructive proof of vizing’s theorem. Inf Process Lett 41(3):131–133
Moret BM (1997) The theory of computation. Addison-Wesley Longman Publishing Co., Inc. Boston, MA
Pudwell L (2008a) Enumeration schemes for pattern-avoiding words and permutations. ProQuest, New Brunswick, New Jersey
Pudwell L (2008b) Enumeration schemes for words avoiding patterns with repeated letters. Electron. J. Comb. Number Theory 8(40):1–19
Schaefer T (1978) The complexity of satisfiability problems. In: Proceedings of the tenth annual ACM symposium on theory of computing, STOC ’78. ACM, New York, NY, USA, pp 216–226
Vizing VG (1964) On an estimate of the chromatic class of a p-graph. J Math Biol 3:23–30 (Russian)
Zhang P (2014) A study on generalized solution concepts in constraint satisfaction and graph colouring. Master’s thesis, University of British Columbia, Canada
Author information
Authors and Affiliations
Corresponding author
Additional information
Parts of this paper were presented at the 21st Annual International Computing and Combinatorics Conference (COCOON 2015), August 4–6, 2015, Beijing, China (Hellmuth and Wieseke 2015). This work was funded by the German Research Foundation (DFG) (Proj. No. MI439/14-1).
Rights and permissions
About this article
Cite this article
Hellmuth, M., Wieseke, N. On tree representations of relations and graphs: symbolic ultrametrics and cograph edge decompositions. J Comb Optim 36, 591–616 (2018). https://doi.org/10.1007/s10878-017-0111-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-017-0111-7