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Hierarchical and Modularly-Minimal Vertex Colorings [article]

Dulce I. Valdivia, Manuela Geiß, Maribel Hernández Rosales, Peter F. Stadler, Marc Hellmuth
2020 arXiv   pre-print
For cographs, the hierarchical colorings coincide with the modularly-minimal coloring.  ...  Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs for which a greedy vertex coloring of every induced subgraph uses only the minimally necessary number of colors χ(G).  ...  Acknowledgments This work was support in part by the German Research Foundation (DFG, STA 850/49-1), the German Federal Ministry of Education and Research (BMBF, project no. 031A538A, de.NBI-RBC), and  ... 
arXiv:2004.06340v1 fatcat:s7e3xz3mpza35blc3wic4oo6f4

Hierarchical and modularly-minimal vertex colorings

Dulce I. Valdivia, Maribel Hernández Rosales, Manuela Geiß, Peter F. Stadler, Marc Hellmuth
2022 The Art of Discrete and Applied Mathematics  
Dedicated to Wilfried Imrich on occasion of his 80th birthday. In a roundabout way, we have to blame Wilfried.  ...  The coloring is a strictly hierarchical and modularly-minimal coloring, see text for further details. of hierarchical colorings and greedy colorings of cographs, and show that a coloring of a cograph is  ...  Every hierarchical coloring of a cograph is also a proper coloring (cf. [9, Lemma 43] ). The property of being a cograph is hereditary (i.e., every induced subgraph of a cograph is a cograph).  ... 
doi:10.26493/2590-9770.1422.9b6 fatcat:5svqkqepk5fdlbrhjp5a3g252m

Transfer flow graphs

Klaus Jansen
1993 Discrete Mathematics  
These graphs G =(iV, E) are generated by nondisjoint union of a cograph and a graph with m disjoint cliques. Such graphs can be described as transfer flow graphs.  ...  We consider four combinatorial optimization problems (independent set, clique, coloring, partition into cliques) for a special graph class.  ...  A subclass of the cographs is given by disjoint union of complete graphs.  ... 
doi:10.1016/0012-365x(93)90488-f fatcat:izicgfulqfav5lgyygvzlnehwe

Defining and identifying cograph communities in complex networks

Songwei Jia, Lin Gao, Yong Gao, James Nastos, Yijie Wang, Xindong Zhang, Haiyang Wang
2015 New Journal of Physics  
the members of the real Arp2/3 complex, and those in the green color are not.  ...  What we want to emphasize is that these meaningful subgroups within cograph communities cannot be detected by the traditional hierarchical algorithms, which focus only on obtaining hierarchical communities  ...  Simple construction of a cotree We describe the simplest polynomial-time algorithm to compute a cotree of a cograph here [9] .  ... 
doi:10.1088/1367-2630/17/1/013044 fatcat:yz6c7ksqvbarjjls2f44arrreu

Processor optimization for flow graphs

Klaus Jansen
1992 Theoretical Computer Science  
First we show that the problem of finding an assignment with minimum number of processors is NP-complete and that the problem of finding a maximum compatible set of operations can be solved in polynomial  ...  In the synthesis of hardware, operations of a scheduled flow graph (acyclic, but with branching nodes) are assigned to processors.  ...  Since the cographs are closed under complement and disjoint union operations, NK, and NK are also cographs. 0 In both cases there exist linear-time algorithms for the coloring problem, see [S, 181. , NK  ... 
doi:10.1016/0304-3975(92)90126-z fatcat:wmdaduapnfhvrdmtekhj6dwtvu

Reciprocal Best Match Graphs [article]

Manuela Geiß and Peter F. Stadler and Marc Hellmuth
2019 arXiv   pre-print
Co-RBMGs are characterized in terms of hierarchically colored cographs, a particular class of vertex colored cographs that is introduced here.  ...  For 3 colors, there are three distinct classes of trees that are related to the structure of the phylogenetic trees explaining them.  ...  Acknowledgements Partial financial support by the German Federal Ministry of Education and Research (BMBF, project no. 031A538A, is gratefully acknowledged.  ... 
arXiv:1903.07920v5 fatcat:n7kgpeajyrhordcl2bh2izvlei

The Mathematics of Xenology: Di-cographs, Symbolic Ultrametrics, 2-structures and Tree-representable Systems of Binary Relations [article]

Marc Hellmuth, Peter F. Stadler, Nicolas Wieseke
2016 arXiv   pre-print
Our main results tie non-symmetric ultrametrics together with di-cographs (the directed generalization of cographs), so-called uniformly non-prime 2-structures, and hierarchical structures on the set of  ...  The corresponding binary relations on a set of genes form complementary cographs. Allowing more than two types of ancestral event types leads to symmetric symbolic ultrametrics.  ...  Colors are labeled with 1,2,3,B (right-most). In fact, all underlying di-graphs G 2 (g), G 3 (g) and G B (g) are di-cographs.  ... 
arXiv:1603.02467v1 fatcat:jrawhmvnbngldbqhku6fwqkmcq

Characterizations of cographs as intersection graphs of paths on a grid

Elad Cohen, Martin Charles Golumbic, Bernard Ries
2014 Discrete Applied Mathematics  
In both cases, we give a characterization by a family of forbidden induced subgraphs. We further present linear-time algorithms to recognize B 1 -EPG cographs and B 0 -VPG cographs using their cotree.  ...  In this paper, we characterize those cographs that are intersection graphs of paths on a grid in the following two cases: (i) the paths on the grid all have at most one bend and the intersections concern  ...  Instead of representing tokens with colors, we use the vector notation of a complete multipartite graph.  ... 
doi:10.1016/j.dam.2014.06.020 fatcat:fufrxu3ka5avba2mckjo5t3qae

From Modular Decomposition Trees to Level-1 Networks: Pseudo-Cographs, Polar-Cats and Prime Polar-Cats [article]

Marc Hellmuth, Guillaume E. Scholz
2022 arXiv   pre-print
In particular, every cograph is a pseudo-cograph and polar prime cats form precisely the class of graphs the can be explained by a labeled level-1 network.  ...  This generalizes the concept of graphs that can be explained by labeled trees, that is, cographs.  ...  Acknowledgments We thank the anonymous designer of the symbol which we downloaded from [13] .  ... 
arXiv:2112.05537v2 fatcat:vlvqi525czasdfwkskksrb4vuy

Linear rankwidth meets stability [article]

Jaroslav Nesetril, Patrice Ossona de Mendez, Roman Rabinovich, Sebastian Siebertz
2019 arXiv   pre-print
family F of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph  ...  Actually, they have bounded c-chromatic number, meaning that they can be colored with f(r) colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary  ...  In particular we thank the reviewers for pointing us to the work of Kwon and Oum [25] .  ... 
arXiv:1911.07748v1 fatcat:mc7ofi4ujjhtfk5ynzci5ly3za

Classes of graphs with low complexity: the case of classes with bounded linear rankwidth [article]

Jaroslav Nesetril, Patrice Ossona de Mendez, Roman Rabinovich, Sebastian Siebertz
2019 arXiv   pre-print
proper hereditary family F of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing  ...  Actually, they have bounded c-chromatic number, meaning that they can be colored with f(r) colors, each color inducing a cograph. 3) To the contrary, based on a Ramsey-like argument, we prove for every  ...  Every graph with linear rankwidth at most can be colored with at most 3( + 2)!2 ( +1 2 ) colors such that each color induces a cograph with cotree height at most + 2.  ... 
arXiv:1909.01564v1 fatcat:tehbh4obbzfjrf5laeavqu3zlq

Author index to volume

2004 Discrete Applied Mathematics  
Sharan, A fully dynamic algorithm for modular decomposition and recognition of cographs (2-3) 329-340 Sharan, R., see R. Shamir (2-3) 329-340 Sikdar, K., see S.  ...  Zhang, Expanding and forwarding parameters of product graphs (1) 63-82 Randerath, B. and I. Schiermeyer, 3-Colorability AP for P 6 -free graphs (2-3) 299-313 Schiermeyer, I., see B.  ... 
doi:10.1016/s0166-218x(03)00669-3 fatcat:nplxrn6oxbdvpf2t3cyzo7wugu

The monadic second-order logic of graphs X: Linear orderings

Bruno Courcelle
1996 Theoretical Computer Science  
Applications concern dependency graphs of partially commutative words, partial k-paths, cographs, and graphs, the modular decomposition of which uses prime graphs of bounded size.  ...  We give some answers to the following general questions: (1) For which classes of graphs % is it possible to specify a linear ordering of the set of vertices of each graph of '8 by fixed monadic second-order  ...  Then y, cannot have color a (because of y3) or b (as brother of yi with color b). Hence it has color c. Hence w has color a.  ... 
doi:10.1016/0304-3975(95)00083-6 fatcat:2vngza3gjbas5odx3lirhesemy

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width [chapter]

B. Courcelle, J. A. Makowsky, U. Rotics
1998 Lecture Notes in Computer Science  
Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones.  ...  We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with "too many" induced paths with four vertices. * J. A.  ...  Acknowledgments We are grateful to Joost Engelfriet for his many valuable comments and suggestions which helped to improve the content of this paper.  ... 
doi:10.1007/10692760_1 fatcat:mi7l6e3kv5g7tk5cxu2h3x66lu

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

B. Courcelle, J. A. Makowsky, U. Rotics
2000 Theory of Computing Systems  
Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones.  ...  We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with "too many" induced paths with four vertices. * J. A.  ...  Acknowledgments We are grateful to Joost Engelfriet for his many valuable comments and suggestions which helped to improve the content of this paper.  ... 
doi:10.1007/s002249910009 fatcat:afveodqdcjdzpa7heteah2jspu
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