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Hereditary equality of domination and exponential domination

Michael .A. Henning, Simon J¤ger, Dieter Rautenbach
2018 Discussiones Mathematicae Graph Theory  
We characterize a large subclass of the class of those graphs G for which the exponential domination number of H equals the domination number of H for every induced subgraph H of G.  ...  Hereditary Equality of Domination and Exponential... 279 Suppose g = 5. This implies that no vertex in R has more than one neighbor on C.  ...  [4] show that computing the exponential domination number is APX-hard for subcubic graphs and describe an efficient algorithm for subcubic trees, but the complexity for general trees is unknown.  ... 
doi:10.7151/dmgt.2006 fatcat:6mzlsf3lprdwhok5pusps3wzq4

Hereditary Equality of Domination and Exponential Domination [article]

Michael A. Henning and Simon Jäger and Dieter Rautenbach
2016 arXiv   pre-print
We characterize a large subclass of the class of those graphs G for which the exponential domination number of H equals the domination number of H for every induced subgraph H of G.  ...  [4] show that the exponential domination number is APX-hard for subcubic graphs and describe an efficient algorithm for subcubic trees, but the complexity for general trees is unknown.  ...  The difficulty to decide whether γ e (G) = γ(G) for a given graph G motivates the study of the hereditary class G of graphs that satisfy this equality, that is, G is the set of those graphs G such that  ... 
arXiv:1605.05056v1 fatcat:ssi5hmqzsnakzc5bj2f26ycfwu

Tight Running Time Lower Bounds for Vertex Deletion Problems [article]

Christian Komusiewicz
2016 arXiv   pre-print
For example, we obtain that Π-Vertex Deletion cannot be solved in 2^o(√(n)) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs.  ...  By a classic result of Lewis and Yannakakis [J. Comput. Syst. Sci. '80], Π-Vertex Deletion is NP-hard for all hereditary properties Π.  ...  Let Π be a hereditary graph property such that Π does not contain all independent sets, then Π-Vertex Deletion cannot be solved in 2 o(n+ √ m) time even if G has a dominating vertex.  ... 
arXiv:1511.05449v2 fatcat:5tuy2fyuyjdjhjs7q4qwp6oclq

Boundary properties of graphs for algorithmic graph problems

Nicholas Korpelainen, Vadim V. Lozin, Dmitriy S. Malyshev, Alexander Tiskin
2011 Theoretical Computer Science  
The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature.  ...  In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and vertex k-colorability.  ...  Acknowledgements The second author acknowledges support of DIMAP -Center for Discrete Mathematics and its Applications at the  ... 
doi:10.1016/j.tcs.2011.03.001 fatcat:bxsjraysjbaejiwfrgjj55jk4e

H-join decomposable graphs and algorithms with runtime single exponential in rankwidth

Binh-Minh Bui-Xuan, Jan Arne Telle, Martin Vatshelle
2010 Discrete Applied Mathematics  
Given an H-join decomposition of an n-vertex m-edge graph G, we solve the Maximum Independent Set and Minimum Dominating Set problems on Rankwidth is a graph parameter introduced by Oum and Seymour, based  ...  the Minimum Dominating Set problem in time O(n(m+2 3 4 k 2 + 23 4 k × k 3 )) and the q-Coloring problem in time O(n(m + 2 q 2 k 2 + 5q+4 2 k × k 2q × q)).  ...  Let T be a subcubic tree and δ a bijection between the leaf set of T and the vertex set of a graph G.  ... 
doi:10.1016/j.dam.2009.09.009 fatcat:i4qashsocbeafocvvwa7ojt33q

Thread Graphs, Linear Rank-Width and Their Algorithmic Applications [chapter]

Robert Ganian
2011 Lecture Notes in Computer Science  
We first provide a characterization of graphs of linear rank-width 1 and then show that on such graphs it is possible to obtain better algorithmic results than on distance hereditary graphs and even trees  ...  Specifically, we provide polynomial algorithms for computing path-width, dominating bandwidth and a 2-approximation of ordinary bandwidth on graphs of linear rank-width 1.  ...  Acknowledgment The author wishes to thank Jan Obdržálek for his contribution and help in characterizing graphs of linear rank-width 1.  ... 
doi:10.1007/978-3-642-19222-7_5 fatcat:luuytuxq2jao7eva5cpygbihqi

Fast FPT algorithms for vertex subset and vertex partitioning problems using neighborhood unions [article]

B.-M. Bui-Xuan and J. A. Telle and M. Vatshelle (Department of Informatics, University of Bergen, Norway)
2011 arXiv   pre-print
Moreover, any optimal rank-decomposition of such a graph will have boolean-width Θ(n), i.e. exponential in the optimal boolean-width.  ...  Next, we exhibit a class of graphs, the Hsu-grids, for which we can solve NP-hard problems in polynomial time, if we use the right parameter.  ...  The graphs of boolean-width at most one are exactly the graphs of rank-width one, i.e. the distance-hereditary graphs.  ... 
arXiv:0903.4796v3 fatcat:vczizctygjd3jobjqpvvjt7vsi

Better algorithms for satisfiability problems for formulas of bounded rank-width [article]

Robert Ganian, Petr Hliněný, Jan Obdržálek
2010 arXiv   pre-print
We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime of which is single-exponential in the rank-width of a formula.  ...  ~[Fischer, Makowsky, and Ravve] -- with a single-exponential dependency on the clique-width of a formula.  ...  hereditary graphs).  ... 
arXiv:1006.5621v1 fatcat:snlmf5aun5bongg3mio7cm4faa

Extension of vertex cover and independent set in some classes of graphs and generalizations [article]

Katrin Casel, Henning Fernau, Mehdi Khosravian Ghadikolaei, Jérôme Monnot, Florian Sikora
2018 arXiv   pre-print
We consider extension variants of the classical graph problems Vertex Cover and Independent Set.  ...  We also study the parameterized complexity of these problems, with parameter |U|, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis.  ...  In conclusion, equality (2) holds and the result follows. For ∆ = 3 3 , we obtain vertex kernel bounds of 4|U | and 2|V \ U |, respectively.  ... 
arXiv:1810.04629v2 fatcat:fbbnrrjkgffepoknyptkjsk5py

Extension of Vertex Cover and Independent Set in Some Classes of Graphs [chapter]

Katrin Casel, Henning Fernau, Mehdi Khosravian Ghadikoalei, Jérôme Monnot, Florian Sikora
2019 Lecture Notes in Computer Science  
We consider extension variants of the classical graph problems Vertex Cover and Independent Set.  ...  We also study the parameterized complexity of theses problems, with parameter |U |, as well as the optimality of simple exact algorithms under the Exponential-Time Hypothesis.  ...  In conclusion, equality (2) holds and the result follows. For ∆ = 3 3 , we obtain vertex kernel bounds of 4|U | and 2|V \ U |, respectively.  ... 
doi:10.1007/978-3-030-17402-6_11 fatcat:xrlkomvp2vbqpbwa33ot2b43ne

Complexity of C_k-Coloring in Hereditary Classes of Graphs

Maria Chudnovsky, Shenwei Huang, Pawel Rzazewski, Sophie Spirkl, Mingxian Zhong, Michael Wagner
2019 European Symposium on Algorithms  
In particular, we consider H-Coloring of F -free graphs, where F is a fixed graph and H is an odd cycle of length at least 5.  ...  For two graphs G and H, an H-coloring of G is a mapping f : V (G) → V (H) such that for every edge uv ∈ E(G) it holds that f (u)f (v) ∈ E(H).  ...  Conclusion In this paper, we initiate a study of C 2s+1 -Coloring for F -free graphs for a fixed graph F .  ... 
doi:10.4230/lipics.esa.2019.31 dblp:conf/esa/ChudnovskyHRSZ19 fatcat:bo5hzlwnjvaozi6ca2ulh4jrae

Enumerating Connected Dominating Sets [article]

Faisal Abu-Khzam, Henning Fernau, Benjamin Gras, Mathieu Liedloff, Kevin Mann
2022 arXiv   pre-print
The question to enumerate all inclusion-minimal connected dominating sets in a graph of order n in time significantly less than 2^n is an open question that was asked in many places.  ...  We also show new lower bound results by constructing a family of graphs of order n with Ω(1.4890^n) minimal connected dominating sets, while previous examples achieved Ω(1.4422^n).  ...  Other improved lower/upper bounds have been obtained for AT-free, strongly chordal, distance-hereditary graphs, and cographs in [20] .  ... 
arXiv:2205.00086v1 fatcat:db75f35k2vdlnjthmegktmcst4

Complexity of C_k-coloring in hereditary classes of graphs [article]

Maria Chudnovsky, Shenwei Huang, Paweł Rzążewski, Sophie Spirkl, Mingxian Zhong
2020 arXiv   pre-print
In particular, we consider H- Coloring of F-free graphs, where F is a fixed graph and H is an odd cycle of length at least 5.  ...  For two graphs G and H, an H-coloring of G is a mapping f:V(G)→ V(H) such that for every edge uv∈ E(G) it holds that f(u)f(v)∈ E(H).  ...  The complexity of k-Coloring for hereditary graph classes has received much attention in the past two decades and significant progress has been made.  ... 
arXiv:2005.01824v1 fatcat:jbrnwhjx2bgapnlqaslwuk2mbi

Twin-width III: Max Independent Set, Min Dominating Set, and Coloring [article]

Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, Rémi Watrigant
2021 arXiv   pre-print
The inevitable price to pay for such a general result is that f is a tower of exponentials of height roughly k. In this paper, we show that algorithms based on twin-width need not be impractical.  ...  We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time f(d,k)n for n-vertex graphs given with a witness that the twin-width is at most d  ...  Due to the Sparsification Lemma [30] and folklore reductions, MIS restricted to subcubic n-vertex graphs cannot be solved in 2 o(n) , under the Exponential Time Hypothesis 13 (ETH) [29] .  ... 
arXiv:2007.14161v2 fatcat:25jkfscxwvbqljzupiftzowacu

On parse trees and Myhill–Nerode-type tools for handling graphs of bounded rank-width

Robert Ganian, Petr Hliněný
2010 Discrete Applied Mathematics  
We illustrate the situation with graph cycles: while C 3 and C 4 have rank-width 1, C 5 and all longer cycles have rank-width equal 2. A rank-decomposition of, say, the cycle C 5 is shown in Fig.  ...  They, moreover, gave new FPT algorithms [4] for solving the independent set, colourability, and dominating set problems on graphs of (bounded) rank-width t in time O(2 Θ(t 2 ) n).  ...  Both authors have been supported by the Research intent MSM0021622419 of the Czech Ministry of Education; Petr Hliněný has moreover been supported by the grant GAČR 201/08/0308 and Robert Ganian by the  ... 
doi:10.1016/j.dam.2009.10.018 fatcat:rcne23i42rbm5n3ehs3f5ikgfu
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