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Linear rank-width and linear clique-width of trees

Isolde Adler, Mamadou Moustapha Kanté
2015 Theoretical Computer Science  
It follows that both 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.  ... 
doi:10.1016/j.tcs.2015.04.021 fatcat:vehu66rrfbfdfkwhnzs75u66km

Linear Rank-Width and Linear Clique-Width of Trees [chapter]

Isolde Adler, Mamadou Moustapha Kanté
2013 Lecture Notes in Computer Science  
It follows that both 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.  ... 
doi:10.1007/978-3-642-45043-3_3 fatcat:sskiv2w6mfcflk7kmvhkw76544

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  ...  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.  ... 
doi:10.1007/978-3-642-19222-7_5 fatcat:luuytuxq2jao7eva5cpygbihqi

Rank-width: Algorithmic and structural results

Sang-il Oum
2017 Discrete Applied Mathematics  
This survey aims to summarize known algorithmic 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.  ... 
doi:10.1016/j.dam.2016.08.006 fatcat:pvvs5yq5hvhxderfshfih3atti

A Note on Graphs of Linear Rank-Width 1 [article]

Binh-Minh Bui-Xuan and Mamadou Moustapha Kanté and Vincent Limouzy
2014 arXiv   pre-print
We prove that a connected graph has 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] .  ... 
arXiv:1306.1345v2 fatcat:2tbpfgma7zebbgprxlapyqxmtm

Comparing Linear Width Parameters for Directed Graphs [article]

Frank Gurski, Carolin Rehs
2018 arXiv   pre-print
In this paper we introduce the 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] .  ... 
arXiv:1812.06653v1 fatcat:6kep7wdvojcgvd7cxbkkjrxd3y

Clique-width and edge contraction [article]

Bruno Courcelle
2013 arXiv   pre-print
We prove that edge contractions do not preserve the property that a set 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.  ... 
arXiv:1306.6168v2 fatcat:5hs4piijp5cx5j37t2irqgg7tu

Clique-width and edge contraction

Bruno Courcelle
2014 Information Processing Letters  
We prove that edge contractions do not preserve the property that a set 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.  ... 
doi:10.1016/j.ipl.2013.09.012 fatcat:fod7pqdk2zbvbgz34lc6lib6tu

Finiteness conditions for graph algebras over tropical semirings [article]

Nadia Labai, Johann A. Makowsky
2014 arXiv   pre-print
Lovász 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  ... 
arXiv:1405.2547v1 fatcat:lwavwoju6vh3tntf5djxkrqu7u

Multi-Clique-Width [article]

Martin Fürer
2015 arXiv   pre-print
In particular, c-colorability can be tested in time 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.  ... 
arXiv:1511.04479v1 fatcat:kjx6vtskffdarbqkknp5vtw2p4

Multi-Clique-Width

Martin Fürer, Marc Herbstritt
2017 Innovations in Theoretical Computer Science  
In particular, c-colorability can be tested in time 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.  ... 
doi:10.4230/lipics.itcs.2017.14 dblp:conf/innovations/Furer17 fatcat:63hodkw3kng3fljbcrtgifjnq4

Tropical Graph Parameters

Nadia Labai, Johann Makowsky
2014 Discrete Mathematics & Theoretical Computer Science  
Lovász 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.  ... 
doi:10.46298/dmtcs.2406 fatcat:uwmvxkb3rra4fiucjod35ol32a

A characterisation of clique-width through nested partitions

Bruno Courcelle, Pinar Heggernes, Daniel Meister, Charis Papadopoulos, Udi Rotics
2015 Discrete Applied Mathematics  
We apply our characterisation to 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  ... 
doi:10.1016/j.dam.2015.02.016 fatcat:j3a3jgbnavfwldxzawpnmb6a7e

Width Parameters Beyond Tree-width and their Applications

P. Hlineny, S.-i. Oum, D. Seese, G. Gottlob
2007 Computer journal  
number 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.  ... 
doi:10.1093/comjnl/bxm052 fatcat:n4nbasuvubc5lkajkskqk22fey

Containment of Monadic Datalog Programs via Bounded Clique-Width [chapter]

Mikołaj Bojańczyk, Filip Murlak, Adam Witkowski
2015 Lecture Notes in Computer Science  
As a step towards these goals, complexity 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  ... 
doi:10.1007/978-3-662-47666-6_34 fatcat:w76f4sqiu5fgljindq67ubzteu
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