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Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs [article]

Jan Bok, Nikola Jedlickova, Barnaby Martin, Pascal Ochem, Daniel Paulusma, Siani Smith
2021 arXiv   pre-print
We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on H-free graphs (for each of the problems, we have one open case  ...  A classical complexity result on Colouring is a well-known dichotomy for H-free graphs (a graph is H-free if it does not contain H as an induced subgraph).  ...  We thank Cyriac Antony for some helpful comments. References  ... 
arXiv:2008.09415v5 fatcat:iihlvmkipvhxrm64tnhv7ylc6m

Acyclic, Star and Injective Colouring: A Complexity Picture for H-Free Graphs

Jan Bok, Nikola Jedlic̆ková, Barnaby Martin, Daniël Paulusma, Siani Smith, Peter Sanders, Fabrizio Grandoni, Grzegorz Herman
2020 European Symposium on Algorithms  
We prove the following results: 1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2) If the number  ...  A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as  ...  We leave a more detailed and systematic complexity study of problems in this framework for future work (see, for example, [11, 23, 24] for some complexity results for both general graphs and special  ... 
doi:10.4230/lipics.esa.2020.22 dblp:conf/esa/BokJMPS20 fatcat:a62lakmnzrggpodmolwr3uicsa

Eighth Workshop and Tutorial on Practical Use of Coloured Petri Nets and the CPN Tools, Aarhus, Denmark, October 22-24, 2007

Kurt Jensen
2007 DAIMI Report Series  
This booklet contains the proceedings of the Eighth Workshop on Practical Use of Coloured Petri Nets and the CPN Tools, October 22-24, 2007.  ...  Acknowledgements We thank the anonymous referees for their careful reading and fruitful comments.  ...  The authors wish to thank the anonymous reviewers for their constructive comments and suggestions that have helped us to improve the CPN model and paper.  ... 
doi:10.7146/dpb.v36i584.7197 fatcat:fmzu5qetpnd6dpflqsuggbt7mm

A discrete Morse perspective on knot projections and a generalised clock theorem [article]

Daniele Celoria, Naya Yerolemou
2021 arXiv   pre-print
Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.  ...  We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on S^2, extending a construction due to Cohen.  ...  If X is a cell complex and Σ = {(x, µ(x))} x∈R an acyclic partial matching on X, then for any pair Y = (y, µ(y)) ∈ Σ, the matching Σ \ Y is acyclic. This follows Figure 8 .  ... 
arXiv:2010.05833v2 fatcat:fbg5tb6smnbeldfseihcmold2i

Filtered simplicial homology, graph dissimilarity and überhomology [article]

Daniele Celoria
2021 arXiv   pre-print
We introduce a natural filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices.  ...  We then use one of these homologies, in the case where X is a graph, to define a conjecturally optimal dissimilarity pseudometric for graphs; we prove various results for this dissimilarity and provide  ...  Figure 4 . 4 A triangulation X of S 2 , exhibiting a dalmatian colouration. The shaded simplices are the closure of the stars of the black vertices. Picture based on [27] .  ... 
arXiv:2105.03987v1 fatcat:cmq64xavwvcf7ajdn6z7i2hnja

Random walk hierarchy measure: What is more hierarchical, a chain, a tree or a star?

Dániel Czégel, Gergely Palla
2015 Scientific Reports  
Although a number of notable methods are already available, their vast majority is treating all directed acyclic graphs as already maximally hierarchical.  ...  Signs of hierarchy are prevalent in a wide range of systems in nature and society.  ...  ., if we set the parameter of the Poisson distribution to α = 0, we obtain b(i) = 1 for all nodes, thus, the resulting graph is actually a chain.  ... 
doi:10.1038/srep17994 pmid:26657012 pmcid:PMC4675073 fatcat:6r5lg642i5afngomeofescotwu

A Discrete Morse Perspective on Knot Projections and a Generalised Clock Theorem

Daniele Celoria, Naya Yerolemou
2021 Electronic Journal of Combinatorics  
Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations.  ...  We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on the two sphere, extending a construction due to Cohen.  ...  The authors would like to thank Vidit Nanda and Agnese Barbensi for their useful feedback and support, and the anonymous referee for their helpful comments.  ... 
doi:10.37236/9979 fatcat:d3pvv5wxkjbcpfrgg27jxwhl7e

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders [article]

Marc Roth, Johannes Schmitt, Philip Wellnitz
2021 arXiv   pre-print
Given a graph property Φ, we consider the problem 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that  ...  We obtain a complete picture for minor-closed properties Φ: the decision problem 𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) always admits an FPT algorithm and the counting problem #𝙴𝚍𝚐𝚎𝚂𝚞𝚋(Φ) always admits an FPTRAS.  ...  Acknowledgements We thank Alina Vdovina and Norbert Peyerimhoff for explaining their construction of 2-group Cayley graph expanders [65] .  ... 
arXiv:2011.03433v2 fatcat:7aauf7xejvfx5atw3n7s6sefsa

On subgroups of Coxeter groups [article]

Warren Dicks, Ian J Leary
2007 arXiv   pre-print
Finally we give a 'bare-hands' construction (using free products with amalgamation and HNN extensions) of a torsion-free group whose integral cohomological dimension is strictly greater than its rational  ...  We also give minimal presentations for certain torsion-free finite-index subgroups of right-angled Coxter groups.  ...  We gratefully acknowledge that this work was generously funded by the DGICYT, through grants PB90-0719 and PB93-0900 for the first-named author, and a post-doctoral fellowship held at the Centre de Recerca  ... 
arXiv:0711.5011v1 fatcat:ijdirw6d4fezdlkqdc7d7s7xr4

Transformations of representation in constraint satisfaction

András Z. Salamon
2015 Constraints  
graphs for which an independent set of specified size can be found efficiently.  ...  I also develop finer grained techniques, by allowing vertices of the microstructure representation to be assigned colours, and the variables to be ordered.  ...  colour class with two vertices (otherwise the graph is completely disconnected), and for the case of a star graph this is all that can be done.  ... 
doi:10.1007/s10601-015-9224-8 fatcat:lollakqzx5h7xeajtfrmaz6u4q

Subject Index Volumes 1–200

2001 Discrete Mathematics  
Erdös and Sós, 4925 Erdös and Szekeres, 4879 Erdös et al., 5529, 5888 Erdös Rubin Taylor, 3815 Erdös Sós Turán, 506 Everett, 2901 Faigle and Sands, 2122 Faudree et al., 4399 Faudree Gould Jacobson  ...  Slater, 5299 Hertz and de Werra, 3748 Hibi, 4839 Hoàng, 3748, 4811 Holton, 1660 Holyer and Cockayne, 1118 Howorka, 2636  ...  C 2n + 1 )-free graph, 6014 bowtie free graph, 6014 H-free graph, 6014 universal cover graph, 3750 universal cycle, 3537, 3740, 4293, 4671 for permutations, 4671 universal cycle of 3-subsets, 3740  ... 
doi:10.1016/s0012-365x(00)00471-4 fatcat:znxg54q6ifgutnhzj3urcw455y

Describing hereditary properties by forbidden circular orderings [article]

Santiago Guzmán-Pro, Pavol Hell, César Hernández-Cruz
2021 arXiv   pre-print
For instance, it is known that a graph G is k-colourable if and only if V(G) admits a linear ordering ≤ with no vertices v_1 ≤⋯≤ v_k+1 such that v_i v_i+1∈ E(G) for every i ∈{ 1, ..., k }.  ...  We show that the classes that can be described by finitely many forbidden circularly ordered graphs include forests, circular-arc graphs, and graphs with circular chromatic number less than k.  ...  Clearly then, f : X → S 1 is an injective function of X into the unit circle and C(f ) = C X . This representation is specially useful when picturing a circular ordering.  ... 
arXiv:2112.00154v1 fatcat:cegehcqevzfwvchulq5y7uelqu

A gentle introduction to Girard's Transcendental Syntax for the linear logician [article]

Boris Eng
2022 arXiv   pre-print
The selected ground for logic in the transcendental syntax is a model of computation I call "stellar resolution" which is basically a logic-free reformulation of Robinson's first-order clausal resolution  ...  All the logical entities and activities will be presented as formatting/structuring on a given model of computation which should be as general, simple and natural as possible.  ...  A cut between A and A ⊥ becomes a binary star [−c.p A (x), −c.p B (x)] coloured with −c for "negative computation". Example 6.  ... 
arXiv:2012.04752v5 fatcat:6lwhknoeafabdjnc5quar5kdtq

Towards Hilbert's 24th Problem: Combinatorial Proof Invariants

Dominic J.D. Hughes
2006 Electronical Notes in Theoretical Computer Science  
This sequel approaches Hilbert's 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces.  ...  Proofs Without Syntax [37] introduced polynomial-time checkable combinatorial proofs for classical propositional logic.  ...  Acknowledgement Many thanks to Rob van Glabbeek for detailed comments on the manuscript. Stanford grant 1DMA644.  ... 
doi:10.1016/j.entcs.2006.05.036 fatcat:2s5awcyzwvho5h2r766j3fmady

Subject index volumes 1–200

1999 Theoretical Computer Science  
850 star-free trace languages, 1806 star-height 1, 278, 3309, 3296 1, regular expressions of -, 278 mjnimal -, 596 star-height of a regular prefix-free language, 596 N-rational series, 3309 star-height  ...  a chessboard-like coloured picture. 3 145 pattern, 2981 tiling recognition, 2764 complexity of -, 2764 tiling systems, 726 the Morse sequence. 1748 with dominoes, 3 145 tilings, 2349, 2423, 2460,2559  ... 
doi:10.1016/s0304-3975(98)00319-3 fatcat:s22ud3iiqjht7lfbtc3zctk7zm
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