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Linear rank-width and linear clique-width of trees

2015
*
Theoretical Computer Science
*

It follows that both

doi:10.1016/j.tcs.2015.04.021
fatcat:vehu66rrfbfdfkwhnzs75u66km
*linear**rank*-*width**and**linear**clique*-*width**of*forests can be computed in*linear*time. ... We show that for every forest T the*linear**rank*-*width**of*T is equal to the path-*width**of*T ,*and*the*linear**clique*-*width**of*T equals the path-*width**of*T plus two, provided that T contains a path*of*length ... We prove Theorem 1*Linear**rank*-*width**and**linear**clique*-*width**of*forests can be computed in*linear*time. ...##
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Linear Rank-Width and Linear Clique-Width of Trees
[chapter]

2013
*
Lecture Notes in Computer Science
*

It follows that both

doi:10.1007/978-3-642-45043-3_3
fatcat:sskiv2w6mfcflk7kmvhkw76544
*linear**rank*-*width**and**linear**clique*-*width**of*forests can be computed in*linear*time. ... We show that for every forest T the*linear**rank*-*width**of*T is equal to the path-*width**of*T ,*and*the*linear**clique*-*width**of*T equals the path-*width**of*T plus two, provided that T contains a path*of*length ... We prove Theorem 1*Linear**rank*-*width**and**linear**clique*-*width**of*forests can be computed in*linear*time. ...##
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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*... In this paper we consider a more restrictive version*of**rank*-*width*called*linear**rank*-*width*, analogously to how pathwidth is obtained from*tree*-*width*. ... 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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Rank-width: Algorithmic and structural results

2017
*
Discrete Applied Mathematics
*

This survey aims to summarize known algorithmic

doi:10.1016/j.dam.2016.08.006
fatcat:pvvs5yq5hvhxderfshfih3atti
*and*structural results on*rank*-*width**of*graphs. ...*Rank*-*width*is a*width*parameter*of*graphs describing whether it is possible to decompose a graph into a*tree*-like structure by 'simple' cuts. ... Kanté, Eun Jung Kim, O-joung Kwon,*and*Jisu Jeong for their helpful comments on an early draft*of*this paper. ...##
###
A Note on Graphs of Linear Rank-Width 1
[article]

2014
*
arXiv
*
pre-print

We prove that a connected graph has

arXiv:1306.1345v2
fatcat:2tbpfgma7zebbgprxlapyqxmtm
*linear**rank*-*width*1 if*and*only if it is a distance-hereditary graph*and*its split decomposition*tree*is a path. ...*rank*-*width*are pivot-minors*of*graphs*of*small*tree*-*width*, arxiv:1203.3606] if*and*only if it does not contain the co-K_2 graph, the Net graph*and*the 5-cycle graph as vertex-minors [Isolde Adler, Arthur ...*Linear**rank*-*width*is related to*rank*-*width*in the same way path-*width*[22] is related to*tree*-*width*[23] . ...##
###
Comparing Linear Width Parameters for Directed Graphs
[article]

2018
*
arXiv
*
pre-print

In this paper we introduce the

arXiv:1812.06653v1
fatcat:6kep7wdvojcgvd7cxbkkjrxd3y
*linear**clique*-*width*,*linear*NLC-*width*, neighbourhood-*width*,*and**linear**rank*-*width*for directed graphs. ... It turns out that the parameters directed*linear**clique*-*width*, directed*linear*NLC-*width*, directed neighbourhood-*width*,*and*directed*linear**rank*-*width*are equivalent in that sense, that all*of*these parameters ... Examples for graph parameters are*tree*-*width*[57] ,*clique*-*width*[17] , NLC-*width*[59] ,*and**rank*-*width*[55] . ...##
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Clique-width and edge contraction
[article]

2013
*
arXiv
*
pre-print

We prove that edge contractions do not preserve the property that a set

arXiv:1306.6168v2
fatcat:5hs4piijp5cx5j37t2irqgg7tu
*of*graphs has bounded*clique*-*width*. ... This property is preserved by contractions*of*edges, one end*of*which is a vertex*of*degree 2. ... Acknowledgement: I thank the anonymous referee who suggested the alternative proof*of*Proposition 1,*and*M. Kanté*and*D. Meister for useful comments. ...##
###
Clique-width and edge contraction

2014
*
Information Processing Letters
*

We prove that edge contractions do not preserve the property that a set

doi:10.1016/j.ipl.2013.09.012
fatcat:fod7pqdk2zbvbgz34lc6lib6tu
*of*graphs has bounded*clique*-*width*. ... Acknowledgement: I thank the anonymous referee who suggested the alternative proof*of*Proposition 1,*and*M. Kanté*and*D. Meister for useful comments. ... A variant*of**clique*-*width*called*linear**clique*-*width*is studied in [3, 10] . We denote by cwd(G)*and*rwd(G) the*clique*-*width**and*, respectively, the*rank*-*width**of*a graph G. ...##
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Finiteness conditions for graph algebras over tropical semirings
[article]

2014
*
arXiv
*
pre-print

Lovász

arXiv:1405.2547v1
fatcat:lwavwoju6vh3tntf5djxkrqu7u
*and*A. Schrijver (2007). Graph parameters with connection matrices*of*finite*rank*can be computed in polynomial time on graph classes*of*bounded*tree*-*width*. ... In the case*of*graph parameters with values in arbitrary commutative semirings, this remains true for graph classes*of*bounded*linear**clique*-*width*. B. Godlin, T. Kotek*and*J.A. ... In [19] the following equivalent definitions*of*the class*of*(labeled or colored) graphs*of**tree*-*width*at most k (TW(k)), path-*width*at most k (PW(k)),*clique*-*width*at most k (CW(k)),*and**linear**clique*-*width*...##
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Multi-Clique-Width
[article]

2015
*
arXiv
*
pre-print

In particular, c-colorability can be tested in time

arXiv:1511.04479v1
fatcat:kjx6vtskffdarbqkknp5vtw2p4
*linear*in n*and*singly exponential in c*and*the*width*k*of*a given multi-k-expression. ... Multi-*clique*-*width*is obtained by a simple modification in the definition*of**clique*-*width*. It has the advantage*of*providing a natural extension*of**tree*-*width*. ...*And*most importantly, there is no explosion*of*the*width*when moving from*tree*-*width*to multi-*clique*-*width*. ...##
###
Multi-Clique-Width

2017
*
Innovations in Theoretical Computer Science
*

In particular, c-colorability can be tested in time

doi:10.4230/lipics.itcs.2017.14
dblp:conf/innovations/Furer17
fatcat:63hodkw3kng3fljbcrtgifjnq4
*linear*in n*and*singly exponential in c*and*the*width*k*of*a given multi-k-expression. ... Multi-*clique*-*width*is obtained by a simple modification in the definition*of**clique*-*width*. It has the advantage*of*providing a natural extension*of**tree*-*width*. ... Any*tree*decomposition*of*bounded*width*can be transformed into a semi-smooth*tree*decomposition in*linear*time. Proof. ...##
###
Tropical Graph Parameters

2014
*
Discrete Mathematics & Theoretical Computer Science
*

Lovász

doi:10.46298/dmtcs.2406
fatcat:uwmvxkb3rra4fiucjod35ol32a
*and*A. Schrijver (2007). Graph parameters with connection matrices*of*finite*rank*can be computed in polynomial time on graph classes*of*bounded*tree*-*width*. ... In the case*of*graph parameters with values in arbitrary commutative semirings, this remains true for graph classes*of*bounded*linear**clique*-*width*. B. Godlin, T. Kotek*and*J.A. ... Our approach also works for • other notions*of**width*for graphs, such as*rank*-*width**and*modular*width*,*and*other inductively defined graph classes, cf. ...##
###
A characterisation of clique-width through nested partitions

2015
*
Discrete Applied Mathematics
*

We apply our characterisation to

doi:10.1016/j.dam.2015.02.016
fatcat:j3a3jgbnavfwldxzawpnmb6a7e
*linear**clique*-*width*. ... We characterise the*clique*-*width*in a combinatorial way by means*of*partitions*of*the vertex set, using*trees**of*nested partitions where partitions are ordered bottom-up by refinement. ...*Tree*-based graph decompositions have*linear*variants: pathwidth is the*linear*variant*of*treewidth,*linear**rank*-*width*is the*linear*variant*of**rank*-*width*,*and**linear**clique*-*width*is the*linear*variant ...##
###
Width Parameters Beyond Tree-width and their Applications

2007
*
Computer journal
*

number

doi:10.1093/comjnl/bxm052
fatcat:n4nbasuvubc5lkajkskqk22fey
*of*other useful parameters like branch-*width*,*rank*-*width*(*clique*-*width*) or hypertree-*width*. ... Besides the very successful concept*of**tree*-*width*(see [Bodlaender, H.*and*Koster, A. (2007) Combinatorial optimisation on graphs*of*bounded treewidth. ... We all thank Jan Obdržálek for giving his expert opinion on digraph*width*notions,*and*J.A. Makowsky*and*the anonymous referee for valuable hints. D.S. thanks T. Dietrich, A. Mitschele*and*S. ...##
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Containment of Monadic Datalog Programs via Bounded Clique-Width
[chapter]

2015
*
Lecture Notes in Computer Science
*

As a step towards these goals, complexity

doi:10.1007/978-3-662-47666-6_34
fatcat:w76f4sqiu5fgljindq67ubzteu
*of*containment over arbitrary structures*of*bounded*clique*-*width*is analysed: satisfiability*and*containment*of*monadic programs with stratified negation is in ... 3-ExpTime,*and*containment*of*a*linear*monadic program in a monadic program is in 2-ExpTime. ... The reason is that, for fixed k, a*tree*decomposition*of**width*k can be computed in*linear*time for graphs*of**tree*-*width*k [4] , but for*clique*-*width*the best currently known polynomial time algorithm ...
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