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Hereditary equality of domination and exponential domination

2018
*
Discussiones Mathematicae Graph Theory
*

We characterize a large subclass

doi:10.7151/dmgt.2006
fatcat:6mzlsf3lprdwhok5pusps3wzq4
*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. ...##
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Hereditary Equality of Domination and Exponential Domination
[article]

2016
*
arXiv
*
pre-print

We characterize a large subclass

arXiv:1605.05056v1
fatcat:ssi5hmqzsnakzc5bj2f26ycfwu
*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 ...##
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Tight Running Time Lower Bounds for Vertex Deletion Problems
[article]

2016
*
arXiv
*
pre-print

For example, we obtain that Π-Vertex Deletion cannot be solved

arXiv:1511.05449v2
fatcat:5tuy2fyuyjdjhjs7q4qwp6oclq
*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. ...##
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Boundary properties of graphs for algorithmic graph problems

2011
*
Theoretical Computer Science
*

The notion

doi:10.1016/j.tcs.2011.03.001
fatcat:bxsjraysjbaejiwfrgjj55jk4e
*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 ...##
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H-join decomposable graphs and algorithms with runtime single exponential in rankwidth

2010
*
Discrete Applied Mathematics
*

Given an H-join decomposition

doi:10.1016/j.dam.2009.09.009
fatcat:i4qashsocbeafocvvwa7ojt33q
*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. ...##
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Thread Graphs, Linear Rank-Width and Their Algorithmic Applications
[chapter]

2011
*
Lecture Notes in Computer Science
*

We first provide a characterization

doi:10.1007/978-3-642-19222-7_5
fatcat:luuytuxq2jao7eva5cpygbihqi
*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. ...##
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Fast FPT algorithms for vertex subset and vertex partitioning problems using neighborhood unions
[article]

2011
*
arXiv
*
pre-print

Moreover, any optimal rank-decomposition

arXiv:0903.4796v3
fatcat:vczizctygjd3jobjqpvvjt7vsi
*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*. ...##
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Better algorithms for satisfiability problems for formulas of bounded rank-width
[article]

2010
*
arXiv
*
pre-print

We provide a parameterized polynomial algorithm for the propositional model counting problem #SAT, the runtime

arXiv:1006.5621v1
fatcat:snlmf5aun5bongg3mio7cm4faa
*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*). ...##
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Extension of vertex cover and independent set in some classes of graphs and generalizations
[article]

2018
*
arXiv
*
pre-print

We consider extension variants

arXiv:1810.04629v2
fatcat:fbbnrrjkgffepoknyptkjsk5py
*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. ...##
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Extension of Vertex Cover and Independent Set in Some Classes of Graphs
[chapter]

2019
*
Lecture Notes in Computer Science
*

We consider extension variants

doi:10.1007/978-3-030-17402-6_11
fatcat:xrlkomvp2vbqpbwa33ot2b43ne
*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. ...##
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Complexity of C_k-Coloring in Hereditary Classes of Graphs

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 . ...

##
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Enumerating Connected Dominating Sets
[article]

2022
*
arXiv
*
pre-print

The question to enumerate all inclusion-minimal connected

arXiv:2205.00086v1
fatcat:db75f35k2vdlnjthmegktmcst4
*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] . ...##
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Complexity of C_k-coloring in hereditary classes of graphs
[article]

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. ...

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

2021
*
arXiv
*
pre-print

The inevitable price to pay for such a general result is that f is a tower

arXiv:2007.14161v2
fatcat:25jkfscxwvbqljzupiftzowacu
*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] . ...##
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On parse trees and Myhill–Nerode-type tools for handling graphs of bounded rank-width

2010
*
Discrete Applied Mathematics
*

We illustrate the situation with

doi:10.1016/j.dam.2009.10.018
fatcat:rcne23i42rbm5n3ehs3f5ikgfu
*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 ...
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