11 Search Results for "Kratochvíl, Jan"


Document
Recognizing H-Graphs - Beyond Circular-Arc Graphs

Authors: Deniz Ağaoğlu Çağırıcı, Onur Çağırıcı, Jan Derbisz, Tim A. Hartmann, Petr Hliněný, Jan Kratochvíl, Tomasz Krawczyk, and Peter Zeman

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


Abstract
In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph H, the class of H-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of H. Such classes of graphs are related to many known graph classes: for example, K₂-graphs coincide with interval graphs, K₃-graphs with circular-arc graphs, the union of T-graphs, where T ranges over all trees, coincides with chordal graphs. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of H-graphs for different graphs H. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborník, and Zeman showed an XP-algorithm testing whether a given graph is a T-graph, where the parameter is the size of the tree T. In particular, for every fixed tree T the recognition of T-graphs can be solved in polynomial time. Tucker showed a polynomial time algorithm recognizing K₃-graphs (circular-arc graphs). On the other hand, Chaplick et al. showed also that for every fixed graph H containing two distinct cycles sharing an edge, the recognition of H-graphs is NP-hard. The main two results of this work narrow the gap between the NP-hard and 𝖯 cases of H-graph recognition. First, we show that the recognition of H-graphs is NP-hard when H contains two distinct cycles. On the other hand, we show a polynomial-time algorithm recognizing L-graphs, where L is a graph containing a cycle and an edge attached to it (which we call lollipop graphs). Our work leaves open the recognition problems of M-graphs for every unicyclic graph M different from a cycle and a lollipop.

Cite as

Deniz Ağaoğlu Çağırıcı, Onur Çağırıcı, Jan Derbisz, Tim A. Hartmann, Petr Hliněný, Jan Kratochvíl, Tomasz Krawczyk, and Peter Zeman. Recognizing H-Graphs - Beyond Circular-Arc Graphs. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 8:1-8:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{agaoglucagirici_et_al:LIPIcs.MFCS.2023.8,
  author =	{A\u{g}ao\u{g}lu \c{C}a\u{g}{\i}r{\i}c{\i}, Deniz and \c{C}a\u{g}{\i}r{\i}c{\i}, Onur and Derbisz, Jan and Hartmann, Tim A. and Hlin\v{e}n\'{y}, Petr and Kratochv{\'\i}l, Jan and Krawczyk, Tomasz and Zeman, Peter},
  title =	{{Recognizing H-Graphs - Beyond Circular-Arc Graphs}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.8},
  URN =		{urn:nbn:de:0030-drops-185420},
  doi =		{10.4230/LIPIcs.MFCS.2023.8},
  annote =	{Keywords: H-graphs, Intersection Graphs, Helly Property}
}
Document
Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases

Authors: Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, and Jan Kratochvíl

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)


Abstract
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs).

Cite as

Jan Bok, Jiří Fiala, Petr Hliněný, Nikola Jedličková, and Jan Kratochvíl. Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{bok_et_al:LIPIcs.MFCS.2021.21,
  author =	{Bok, Jan and Fiala, Ji\v{r}{\'\i} and Hlin\v{e}n\'{y}, Petr and Jedli\v{c}kov\'{a}, Nikola and Kratochv{\'\i}l, Jan},
  title =	{{Computational Complexity of Covering Multigraphs with Semi-Edges: Small Cases}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.21},
  URN =		{urn:nbn:de:0030-drops-144611},
  doi =		{10.4230/LIPIcs.MFCS.2021.21},
  annote =	{Keywords: graph cover, covering projection, semi-edges, multigraphs, complexity}
}
Document
U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

Authors: Jan Kratochvíl, Tomáš Masařík, and Jana Novotná

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)


Abstract
Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Cite as

Jan Kratochvíl, Tomáš Masařík, and Jana Novotná. U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 57:1-57:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kratochvil_et_al:LIPIcs.MFCS.2020.57,
  author =	{Kratochv{\'\i}l, Jan and Masa\v{r}{\'\i}k, Tom\'{a}\v{s} and Novotn\'{a}, Jana},
  title =	{{U-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{57:1--57:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.57},
  URN =		{urn:nbn:de:0030-drops-127247},
  doi =		{10.4230/LIPIcs.MFCS.2020.57},
  annote =	{Keywords: Interval Graphs, Mixed Unit Interval Graphs, MaxCut Problem, Clique Width, Subexponential Algorithm, Bubble Model}
}
Document
Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties

Authors: Alberto Dennunzio, Enrico Formenti, Darij Grinberg, and Luciano Margara

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e. , with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case.

Cite as

Alberto Dennunzio, Enrico Formenti, Darij Grinberg, and Luciano Margara. Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{dennunzio_et_al:LIPIcs.MFCS.2019.68,
  author =	{Dennunzio, Alberto and Formenti, Enrico and Grinberg, Darij and Margara, Luciano},
  title =	{{Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{68:1--68:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.68},
  URN =		{urn:nbn:de:0030-drops-110126},
  doi =		{10.4230/LIPIcs.MFCS.2019.68},
  annote =	{Keywords: Cellular Automata, Symbolic Dynamics, Complex Systems}
}
Document
A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs

Authors: Lars Jaffke and Paloma T. Lima

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has at least one neighbor in all the other color classes. The b-Coloring problem asks whether a graph G has a b-coloring with k colors. The b-chromatic number of a graph G, denoted by chi_b(G), is the maximum number k such that G admits a b-coloring with k colors. We consider the complexity of the b-Coloring problem, whenever the value of k is close to one of two upper bounds on chi_b(G): The maximum degree Delta(G) plus one, and the m-degree, denoted by m(G), which is defined as the maximum number i such that G has i vertices of degree at least i-1. We obtain a dichotomy result for all fixed k in N when k is close to one of the two above mentioned upper bounds. Concretely, we show that if k in {Delta(G) + 1 - p, m(G) - p}, the problem is polynomial-time solvable whenever p in {0, 1} and, even when k = 3, it is NP-complete whenever p >= 2. We furthermore consider parameterizations of the b-Coloring problem that involve the maximum degree Delta(G) of the input graph G and give two FPT-algorithms. First, we show that deciding whether a graph G has a b-coloring with m(G) colors is FPT parameterized by Delta(G). Second, we show that b-Coloring{} is FPT parameterized by Delta(G) + l_k(G), where l_k(G) denotes the number of vertices of degree at least k.

Cite as

Lars Jaffke and Paloma T. Lima. A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 34:1-34:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{jaffke_et_al:LIPIcs.MFCS.2019.34,
  author =	{Jaffke, Lars and Lima, Paloma T.},
  title =	{{A Complexity Dichotomy for Critical Values of the b-Chromatic Number of Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{34:1--34:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.34},
  URN =		{urn:nbn:de:0030-drops-109784},
  doi =		{10.4230/LIPIcs.MFCS.2019.34},
  annote =	{Keywords: b-Coloring, b-Chromatic Number}
}
Document
On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond

Authors: Paul C. Bell, Igor Potapov, and Pavel Semukhin

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
We consider the following variant of the Mortality Problem: given k x k matrices A_1, A_2, ...,A_{t}, does there exist nonnegative integers m_1, m_2, ...,m_t such that the product A_1^{m_1} A_2^{m_2} * ... * A_{t}^{m_{t}} is equal to the zero matrix? It is known that this problem is decidable when t <= 2 for matrices over algebraic numbers but becomes undecidable for sufficiently large t and k even for integral matrices. In this paper, we prove the first decidability results for t>2. We show as one of our central results that for t=3 this problem in any dimension is Turing equivalent to the well-known Skolem problem for linear recurrence sequences. Our proof relies on the Primary Decomposition Theorem for matrices that was not used to show decidability results in matrix semigroups before. As a corollary we obtain that the above problem is decidable for t=3 and k <= 3 for matrices over algebraic numbers and for t=3 and k=4 for matrices over real algebraic numbers. Another consequence is that the set of triples (m_1,m_2,m_3) for which the equation A_1^{m_1} A_2^{m_2} A_3^{m_3} equals the zero matrix is equal to a finite union of direct products of semilinear sets. For t=4 we show that the solution set can be non-semilinear, and thus it seems unlikely that there is a direct connection to the Skolem problem. However we prove that the problem is still decidable for upper-triangular 2 x 2 rational matrices by employing powerful tools from transcendence theory such as Baker’s theorem and S-unit equations.

Cite as

Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bell_et_al:LIPIcs.MFCS.2019.83,
  author =	{Bell, Paul C. and Potapov, Igor and Semukhin, Pavel},
  title =	{{On the Mortality Problem: From Multiplicative Matrix Equations to Linear Recurrence Sequences and Beyond}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{83:1--83:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.83},
  URN =		{urn:nbn:de:0030-drops-110279},
  doi =		{10.4230/LIPIcs.MFCS.2019.83},
  annote =	{Keywords: Linear recurrence sequences, Skolem’s problem, mortality problem, matrix equations, primary decomposition theorem, Baker’s theorem}
}
Document
Colouring H-Free Graphs of Bounded Diameter

Authors: Barnaby Martin, Daniël Paulusma, and Siani Smith

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
The Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for an integer k, such that no two adjacent vertices are coloured alike. A graph G is H-free if G does not contain H as an induced subgraph. It is known that Colouring is NP-complete for H-free graphs if H contains a cycle or claw, even for fixed k >= 3. We examine to what extent the situation may change if in addition the input graph has bounded diameter.

Cite as

Barnaby Martin, Daniël Paulusma, and Siani Smith. Colouring H-Free Graphs of Bounded Diameter. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 14:1-14:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{martin_et_al:LIPIcs.MFCS.2019.14,
  author =	{Martin, Barnaby and Paulusma, Dani\"{e}l and Smith, Siani},
  title =	{{Colouring H-Free Graphs of Bounded Diameter}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{14:1--14:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.14},
  URN =		{urn:nbn:de:0030-drops-109584},
  doi =		{10.4230/LIPIcs.MFCS.2019.14},
  annote =	{Keywords: vertex colouring, H-free graph, diameter}
}
Document
Parameterized Algorithms for Deletion to (r,ell)-Graphs

Authors: Sudeshna Kolay and Fahad Panolan

Published in: LIPIcs, Volume 45, 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)


Abstract
For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

Cite as

Sudeshna Kolay and Fahad Panolan. Parameterized Algorithms for Deletion to (r,ell)-Graphs. In 35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 45, pp. 420-433, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{kolay_et_al:LIPIcs.FSTTCS.2015.420,
  author =	{Kolay, Sudeshna and Panolan, Fahad},
  title =	{{Parameterized Algorithms for Deletion to (r,ell)-Graphs}},
  booktitle =	{35th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2015)},
  pages =	{420--433},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-97-2},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{45},
  editor =	{Harsha, Prahladh and Ramalingam, G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2015.420},
  URN =		{urn:nbn:de:0030-drops-56456},
  doi =		{10.4230/LIPIcs.FSTTCS.2015.420},
  annote =	{Keywords: FPT, Turing kernels, Approximation algorithms}
}
Document
B-Chromatic Number: Beyond NP-Hardness

Authors: Fahad Panolan, Geevarghese Philip, and Saket Saurabh

Published in: LIPIcs, Volume 43, 10th International Symposium on Parameterized and Exact Computation (IPEC 2015)


Abstract
The b-chromatic number of a graph G, chi_b(G), is the largest integer k such that G has a k-vertex coloring with the property that each color class has a vertex which is adjacent to at least one vertex in each of the other color classes. In the B-Chromatic Number problem, the objective is to decide whether chi_b(G) >= k. Testing whether chi_b(G)=Delta(G)+1, where Delta(G) is the maximum degree of a graph, itself is NP-complete even for connected bipartite graphs (Kratochvil, Tuza and Voigt, WG 2002). In this paper we study B-Chromatic Number in the realm of parameterized complexity and exact exponential time algorithms. We show that B-Chromatic Number is W[1]-hard when parameterized by k, resolving the open question posed by Havet and Sampaio (Algorithmica 2013). When k=Delta(G)+1, we design an algorithm for B-Chromatic Number running in time 2^{O(k^2 * log(k))}*n^{O(1)}. Finally, we show that B-Chromatic Number for an n-vertex graph can be solved in time O(3^n * n^{4} * log(n)).

Cite as

Fahad Panolan, Geevarghese Philip, and Saket Saurabh. B-Chromatic Number: Beyond NP-Hardness. In 10th International Symposium on Parameterized and Exact Computation (IPEC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 43, pp. 389-401, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{panolan_et_al:LIPIcs.IPEC.2015.389,
  author =	{Panolan, Fahad and Philip, Geevarghese and Saurabh, Saket},
  title =	{{B-Chromatic Number: Beyond NP-Hardness}},
  booktitle =	{10th International Symposium on Parameterized and Exact Computation (IPEC 2015)},
  pages =	{389--401},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-92-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{43},
  editor =	{Husfeldt, Thore and Kanj, Iyad},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2015.389},
  URN =		{urn:nbn:de:0030-drops-55997},
  doi =		{10.4230/LIPIcs.IPEC.2015.389},
  annote =	{Keywords: b-chromatic number, exact algorithm, parameterized complexity}
}
Document
Recognition and Complexity of Point Visibility Graphs

Authors: Jean Cardinal and Udo Hoffmann

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.

Cite as

Jean Cardinal and Udo Hoffmann. Recognition and Complexity of Point Visibility Graphs. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 171-185, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{cardinal_et_al:LIPIcs.SOCG.2015.171,
  author =	{Cardinal, Jean and Hoffmann, Udo},
  title =	{{Recognition and Complexity of Point Visibility Graphs}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{171--185},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.171},
  URN =		{urn:nbn:de:0030-drops-51390},
  doi =		{10.4230/LIPIcs.SOCG.2015.171},
  annote =	{Keywords: point visibility graphs, recognition, existential theory of the reals}
}
Document
Towards an Isomorphism Dichotomy for Hereditary Graph Classes

Authors: Pascal Schweitzer

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)


Abstract
In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color class size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.

Cite as

Pascal Schweitzer. Towards an Isomorphism Dichotomy for Hereditary Graph Classes. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 689-702, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{schweitzer:LIPIcs.STACS.2015.689,
  author =	{Schweitzer, Pascal},
  title =	{{Towards an Isomorphism Dichotomy for Hereditary Graph Classes}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{689--702},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.689},
  URN =		{urn:nbn:de:0030-drops-49513},
  doi =		{10.4230/LIPIcs.STACS.2015.689},
  annote =	{Keywords: graph isomorphism, modular decomposition, bounded color valence, reductions, forbidden induced subgraphs}
}
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