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Hierarchical and Modularly-Minimal Vertex Colorings
[article]
2020
arXiv
pre-print
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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
2022
The Art of Discrete and Applied Mathematics
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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
1993
Discrete Mathematics
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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
2015
New Journal of Physics
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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
1992
Theoretical Computer Science
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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]
2019
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
2014
Discrete Applied Mathematics
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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]
2022
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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
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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
1996
Theoretical Computer Science
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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]
1998
Lecture Notes in Computer Science
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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
2000
Theory of Computing Systems
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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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