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On the tree-width of even-hole-free graphs
[article]

2020
*
arXiv
*
pre-print

We prove

arXiv:2008.05504v1
fatcat:ij2smflzl5dx5h6sladobes5ve
*the*conjecture for subcubic*graphs*and we give a bound*on**the**tree*-*width**of**the*class*of*(*even**hole*, pyramid)-*free**graphs**of*degree at most 4. ...*The*class*of*all*even*-*hole*-*free**graphs*has unbounded*tree*-*width*, as it contains all complete*graphs*. ...*The*class*of*all*even*-*hole*-*free**graphs*trivially has unbounded*tree*-*width*, as is contains all complete*graphs*. ...##
###
On rank-width of even-hole-free graphs
[article]

2017
*
arXiv
*
pre-print

We present a class

arXiv:1611.09907v2
fatcat:hgtsktipz5gytac7dkcjlz62le
*of*(diamond,*even**hole*)-*free**graphs*with no clique cutset that has unbounded rank-*width*. ... In general,*even*-*hole*-*free**graphs*have unbounded rank-*width*, because chordal*graphs*are*even*-*hole*-*free*. A.A. da Silva, A. Silva and C. ... Rank-*width**of*subclasses*of**even*-*hole*-*free**graphs*has also been studied. In [12] it is shown that planar*even*-*hole*-*free**graphs*have*tree*-*width*at most 49. ...##
###
Coloring even-hole-free graphs with no star cutset
[article]

2018
*
arXiv
*
pre-print

In this paper, we study

arXiv:1805.01948v1
fatcat:ro44af5s6zaetefh4ewn37qqju
*the*class*of**even*-*hole*-*free**graphs*with no star cutset. ... A*hole*is a chordless cycle*of*length at least 4. A*graph*is*even*-*hole*-*free*if it does not contain any*hole**of**even*length as an induced subgraph. ...*The*rank-*width**of*a clique is at most 1 and*the*rank-*width**of*a*hole*is at most 2. 3 χ-bounding function 3 .1 Special*graphs*Recall that*the*bound*of*chromatic number for*even*-*hole*-*free**graphs*(χ(G) ...##
###
On the structure of (pan, even hole)-free graphs
[article]

2017
*
arXiv
*
pre-print

Using this structure theorem, we show that

arXiv:1508.03062v2
fatcat:futyzwrk6zhp3gxxtreiof2cti
*the**tree*-*width**of*a (pan,*even**hole*)-*free**graph*is at most 1.5 times*the*clique number minus 1, and thus*the*chromatic number is at most 1.5 times*the*clique ... A*hole*is a chordless cycle with at least four vertices. A pan is a*graph*which consists*of*a*hole*and a single vertex with precisely*one*neighbor*on**the**hole*. ... Problem 7.2 Is*the**tree*-*width**of*an*even*-*hole*-*free**graph*bounded by a function*of*its clique number? ...##
###
Structure and algorithms for (cap, even hole)-free graphs
[article]

2016
*
arXiv
*
pre-print

A

arXiv:1611.08066v1
fatcat:we3h6o6bofeldlbzy6ouyaux6y
*graph*is*even*-*hole*-*free*if it has no induced*even*cycles*of*length 4 or more. A cap is a cycle*of*length at least 5 with exactly*one*chord and that chord creates a triangle with*the*cycle. ... In this paper, we consider (cap,*even**hole*)-*free**graphs*, and more generally, (cap, 4-*hole*)-*free*odd-signable*graphs*. We give an explicit construction*of*these*graphs*. ... We would like to thank Haiko Müller for fruitful discussions, and to thank Jerry Spinrad for pointing to us*the*idea*of*finding true twins in O(n + m) time. ...##
###
Structure and algorithms for (cap, even hole)-free graphs

2018
*
Discrete Mathematics
*

Abstract A

doi:10.1016/j.disc.2017.09.013
fatcat:lrcmtmfz4rg5hd6iikquqhpnny
*graph*is*even*-*hole*-*free*if it has no induced*even*cycles*of*length 4 or more. A cap is a cycle*of*length at least 5 with exactly*one*chord and that chord creates a triangle with*the*cycle. ... In this paper, we consider (cap,*even**hole*)-*free**graphs*, and more generally, (cap, 4-*hole*)-*free*odd-signable*graphs*. We give an explicit construction*of*these*graphs*. ... We would like to thank Haiko Müller for fruitful discussions, and to thank Jerry Spinrad for pointing to us*the*idea*of*finding true twins in O(n + m) time. ...##
###
Page 779 of Mathematical Reviews Vol. , Issue 2001B
[page]

2001
*
Mathematical Reviews
*

David Burns (1-FERR; Big Rapids, MI)
2001b:05189 05C75
Kotlov, Andrei (3-WTRL-B; Waterloo,

*ON*) Spectral characterization*of**tree*-*width*-two*graphs*. ... A subset*of*edges in a signed*graph*is odd [*even*] if it contains an odd [*even*] number*of*odd labeled edges. A*hole*in a*graph*is a chordless cycle containing at least four edges. ...##
###
Transformations of representation in constraint satisfaction

2015
*
Constraints
*

My main contribution is an investigation

doi:10.1007/s10601-015-9224-8
fatcat:lollakqzx5h7xeajtfrmaz6u4q
*of*such hybrid CSPs which cannot be defined with either*one**of*these kinds*of*restrictions. ...*graphs*for which an independent set*of*specified size can be found efficiently. ... Since MS(*Tree*) is hereditary, it is therefore*the*class*of*(X, odd-*hole*, antihole)-*free**graphs*, for some set X*of**graphs*. ...##
###
From χ- to χ_p-bounded classes
[article]

2021
*
arXiv
*
pre-print

As an application

arXiv:2009.02953v2
fatcat:woph6rwtvbdg7p3yjp5xxnsodq
*of*our characterizations, among other things, we show that for every odd integer g>3*even**hole*-*free**graphs*G contain at most φ(g,ω(G)) |G|*holes**of*length g. ...*On**the*way, we generalize a result*of*Wood relating*the*chromatic number*of*a*graph*to*the*star chromatic number*of*its 1-subdivision. ... Acknowledgements*The*authors would like to thank*the*anonymous referees for his corrections and suggestions. ...##
###
(Theta, triangle)-free and (even hole, K_4)-free graphs. Part 1 : Layered wheels
[article]

2021
*
arXiv
*
pre-print

They also provide examples

arXiv:1906.10998v5
fatcat:fet7nsotbzcvflkd7huybdfomm
*of**graphs**of*large treewidth and large rankwidth in well-studied classes, such as (theta, triangle)-*free**graphs*and*even*-*hole*-*free**graphs*with no K_4 (where a*hole*is a chordless ... cycle*of*length at least four, a theta is a*graph*made*of*three internally vertex disjoint paths*of*length at least two linking two vertices, and K_4 is*the*complete*graph**on*four vertices). ...*Even*-*hole*-*free**graphs*Our last motivation for this work is a better understanding*of**even*-*hole*-*free**graphs*. ...##
###
The independent set problem is FPT for even-hole-free graphs
[article]

2019
*
arXiv
*
pre-print

*The*class

*of*

*even*-

*hole*-

*free*

*graphs*is very similar to

*the*class

*of*perfect

*graphs*, and was indeed a cornerstone in

*the*tools leading to

*the*proof

*of*

*the*Strong Perfect

*Graph*Theorem. ... Halfway

*of*these, we show in this paper that MIS is FPT when parameterized by

*the*solution size in

*the*class

*of*

*even*-

*hole*-

*free*

*graphs*. ... Closely related to

*the*class

*of*perfect

*graphs*is

*the*class

*of*

*even*-

*hole*-

*free*

*graphs*.

*The*class

*of*

*even*-

*hole*-

*free*

*graphs*was introduced as a class structurally similar to

*the*class

*of*Berge

*graphs*. ...

##
###
The Independent Set Problem Is FPT for Even-Hole-Free Graphs

2019
*
International Symposium on Parameterized and Exact Computation
*

*The*class

*of*

*even*-

*hole*-

*free*

*graphs*is very similar to

*the*class

*of*perfect

*graphs*, and was indeed a cornerstone in

*the*tools leading to

*the*proof

*of*

*the*Strong Perfect

*Graph*Theorem. ... Halfway

*of*these, we show in this paper that MIS is FPT when parameterized by

*the*solution size in

*the*class

*of*

*even*-

*hole*-

*free*

*graphs*. ... Lemma 8 shows that

*the*bi-

*trees*arising from

*even*-

*hole*-

*free*

*graphs*do not have

*the*obstructions. ...

##
###
A note on module-composed graphs
[article]

2007
*
arXiv
*
pre-print

In this paper we consider module-composed

arXiv:0705.1521v2
fatcat:sid2wof7znhcnlhdj7lk35zrfy
*graphs*, i.e.*graphs*which can be defined by a sequence*of**one*-vertex insertions v_1,... ... ,v_n, such that*the*neighbourhood*of*vertex v_i, 2<= i<= n, forms a module (a homogeneous set)*of**the**graph*defined by vertices v_1,..., v_i-1. ... Examples for distance hereditary*graphs*are co-*graphs*and*trees*. For*the*case*of*bipartite*graphs*8 ,*the*notion module-composed*even*is equivalent to*the*notion*of*distance hereditary. 1. ...##
###
Special tree-width and the verification of monadic second-order graph pr operties

2010
*
Foundations of Software Technology and Theoretical Computer Science
*

*The*model-checking problem for monadic second-order logic

*on*

*graphs*is fixed-parameter tractable with respect to

*tree*-

*width*and clique-

*width*. ... We introduce special

*tree*-

*width*, a

*graph*complexity measure between path-

*width*and

*tree*-

*width*.

*The*corresponding automata are easier to construct than those for

*tree*-

*width*. ... It appears that

*the*corresponding automata,

*even*for

*the*basic property

*of*adjacency, have exponential size in

*the*bound

*on*

*tree*-

*width*. ...

##
###
Surfaces, Tree-Width, Clique-Minors, and Partitions

2000
*
Journal of combinatorial theory. Series B (Print)
*

In 1971, Chartrand, Geller, and Hedetniemi conjectured that

doi:10.1006/jctb.2000.1962
fatcat:j7hw2spwajeaparmwn4uo4voae
*the*edge set*of*a planar*graph*may be partitioned into two subsets, each*of*which induces an outerplanar*graph*. ...*One*such result, in which a planar*graph*may be thus edge partitioned into two series-parallel*graphs*, has nice generalizations for*graphs*embedded onto an arbitrary surface and*graphs*with no large clique-minor ...*The**tree*-*width**of*a*graph*G, in symbols tw(G), is*the*minimum integer k such that G is a partial k-*tree*. Another important class*of**graphs*are*graphs*embeddable*on*a given surface. ...
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