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Hereditary equality of domination and exponential domination
2018
Discussiones Mathematicae Graph Theory
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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]
2016
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
2011
Theoretical Computer Science
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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
2010
Discrete Applied Mathematics
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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]
2011
Lecture Notes in Computer Science
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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]
2011
arXiv
pre-print
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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]
2010
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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]
2019
Lecture Notes in Computer Science
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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
2019
European Symposium on Algorithms
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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]
2022
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
2010
Discrete Applied Mathematics
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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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