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Well Quasi Orders in Subclasses of Bounded Treewidth Graphs and Their Algorithmic Applications

2011
*
Algorithmica
*

We show that three

doi:10.1007/s00453-011-9545-y
fatcat:cwobkdbqxnbwtnnc2njzm2lnci
*subclasses**of**bounded**treewidth**graphs*are*well*-*quasi*-*ordered*by refinements*of*the minor*order*. ... Specifically, we prove that*graphs*with*bounded*vertex-covers are*well**quasi**ordered*by the induced subgraph*order*,*graphs*with*bounded*feedback-vertex-set are*well**quasi**ordered*by the topological-minor ... Our main result shows that nevertheless these*orders*are*in*fact*well*-*quasi*-*orders*on some natural*subclasses**of**bounded**treewidth**graphs*. ...##
###
Guest Editorial: Special Issue on Parameterized and Exact Computation, Part I

2011
*
Algorithmica
*

The paper '

doi:10.1007/s00453-011-9605-3
fatcat:5bwu4ykmlrcypdm5wzdvtehirq
*Well**Quasi**orders**in**subclasses**of**bounded**treewidth*... The size*of*the vertex cover (which measures how far the*graph*is from an edgeless*graph*) and the feedback vertex set (which measures how far the*graph*is from a forest) are such parameters especially ... The paper '*Well**Quasi**orders**in**subclasses**of**bounded**treewidth**graphs*and their algorithmic applications' revisits the classical Robertson-Seymour theorem*in*the classes*of**bounded**treewidth**graphs*that ...##
###
Fast Diameter Computation within Split Graphs
[article]

2021
*
arXiv
*
pre-print

We prove that it is the case for k=1 and for some

arXiv:1910.03438v5
fatcat:kfmwqqhqe5a7blrimrt6qzgihq
*subclasses*such as*bounded*-*treewidth*split*graphs*, threshold*graphs*and comparability split*graphs*. ... When can we compute the diameter*of*a*graph**in**quasi*linear time? ...*In*Sec. 4 we prove that it is indeed the case for*bounded*-*treewidth*split*graphs*(trivially) and some other dense*subclasses**of**bounded*clique-interval number such as comparability split*graphs*. ...##
###
Fast Diameter Computation within Split Graphs

2021
*
Discrete Mathematics & Theoretical Computer Science
*

We prove that it is the case for $k=1$ and for some

doi:10.46298/dmtcs.6422
fatcat:qr7voo4ic5aaddhqnd7n3w5xl4
*subclasses*such as*bounded*-*treewidth*split*graphs*, threshold*graphs*and comparability split*graphs*. ... When can we compute the diameter*of*a*graph**in**quasi*linear time? ...*In*Sec. 4 we prove that it is indeed the case for*bounded*-*treewidth*split*graphs*(trivially) and some other dense*subclasses**of**bounded*clique-interval number such as comparability split*graphs*. ...##
###
Clique-Width for Hereditary Graph Classes
[article]

2019
*
arXiv
*
pre-print

We also explain a possible strong connection between results on

arXiv:1901.00335v2
fatcat:3mb44qloc5cxxddyimwo6horqa
*boundedness**of*clique-width and on*well*-*quasi*-*orderability*by the induced subgraph relation for hereditary*graph*classes. ... Clique-width is a*well*-studied*graph*parameter owing to its use*in*understanding algorithmic tractability: if the clique-width*of*a*graph*class G is*bounded*by a constant, a wide range*of*problems that ... We now present the state-*of*-the-art summary for*well*-*quasi*-*orderability*for classes on (H 1 , H 2 )-free*graphs*, which is obtained by combining results from [6, 63, 64, 72, 122, 123] . ...##
###
Beyond Outerplanarity
[chapter]

2018
*
Lecture Notes in Computer Science
*

Thus, since outer k-planar

doi:10.1007/978-3-319-73915-1_42
fatcat:r523bjfdojbotfnm7pxuzkmvry
*graphs*have*bounded**treewidth*, closed outer k-planarity is linear-time testable by Courcelle's Theorem. ... For each k, we express closed outer k-planarity and closed outer k-*quasi*-planarity*in*extended monadic second-*order*logic. ...*Graphs*with*bounded**treewidth*are*well*-known due to Courcelle's Theorem (see Theorem 6) [10] , i.e., having*bounded**treewidth*means many problems can be solved efficiently. ...##
###
Graphs without large bicliques and well-quasi-orderability by the induced subgraph relation
[article]

2015
*
arXiv
*
pre-print

Recently, Daligault, Rao and Thomass\'e asked

arXiv:1410.3260v2
fatcat:kjo6xlckn5hjdiegokkvbjkhoa
*in*[3] if every hereditary class which is*well*-*quasi*-*ordered*by the induced subgraph relation is*of**bounded*clique-width. ...*In*particular, this will mean (through the use*of*Courcelle theorem [2]), that any problem definable*in*Monadic Second*Order*Logic can be solved*in*a polynomial time on any class*well*-*quasi*-*ordered*by ... If X is a hereditary*subclass**of*(K t , K q,q )-free*graphs*which is*well*-*quasi*-*ordered*by the induced subgraph relation, then X has a*bounded**treewidth*. ...##
###
Beyond Outerplanarity
[article]

2017
*
arXiv
*
pre-print

For each k, we express closed outer k-planarity and closed outer k-

arXiv:1708.08723v2
fatcat:4bj7qcxbyfdspj4wxlinj3vs5q
*quasi*-planarity*in*extended monadic second-*order*logic. ... We study straight-line drawings*of**graphs*where the vertices are placed*in*convex position*in*the plane, i.e., convex drawings. ...*Graphs*with*bounded**treewidth*are*well*-known due to Courcelle's Theorem (see Theorem 6) [10] , i.e., having*bounded**treewidth*means many problems can be solved efficiently. ...##
###
On the read-once property of branching programs and CNFs of bounded treewidth
[article]

2015
*
arXiv
*
pre-print

*In*this paper we prove a space lower

*bound*

*of*n^Ω(k) for non-deterministic (syntactic) read-once branching programs ( nrobps) on functions expressible as cnfs with

*treewidth*at most k

*of*their primal

*graphs*... We use lower

*bound*for nrobps to obtain a

*quasi*-polynomial separation between Free Binary Decision Diagrams and Decision Decomposable Negation Normal Forms, essentially matching the existing upper

*bound*... Oblivious ROBPs, better known as

*Ordered*Binary Decision Diagrams (OBDDs), are a

*subclass*

*of*ROBPs, very

*well*known because

*of*its applications

*in*the area

*of*verification [3] . ...

##
###
On the Read-Once Property of Branching Programs and CNFs of Bounded Treewidth

2015
*
Algorithmica
*

*In*this paper we prove a space lower

*bound*

*of*n Ω(k) for non-deterministic (syntactic) read-once branching programs (NROBPs) on functions expressible as CNFs with

*treewidth*at most k

*of*their primal

*graphs*... We use lower

*bound*for NROBPs to obtain a

*quasi*-polynomial separation between Free Binary Decision Diagrams and Decision Decomposable Negation Normal Forms, essentially matching the existing upper

*bound*... Oblivious ROBPs, better known as

*Ordered*Binary Decision Diagrams (OBDDs), are a

*subclass*

*of*ROBPs, very

*well*known because

*of*its applications

*in*the area

*of*verification [3] . ...

##
###
Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes

2017
*
Order
*

Daligault, Rao and Thomassé asked whether a hereditary class

doi:10.1007/s11083-017-9430-7
fatcat:setirbra4jemnp4wbxrgdcuo44
*of**graphs**well*-*quasi*-*ordered*by the induced subgraph relation has*bounded*clique-width. ... The conjecture is known to hold for classes*of**graphs*defined by a single forbidden induced subgraph H, as such*graphs*are*well*-*quasi*-*ordered*and are*of**bounded*clique-width if and only if H is an induced ... One*of*the first steps towards this result was the proof*of*the fact that*graph*classes*of**bounded**treewidth*are*well*-*quasi*-*ordered*by the minor relation [26] (a*graph*parameter π is said to be*bounded*...##
###
Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes
[chapter]

2016
*
Lecture Notes in Computer Science
*

*In*this case, the two notions even coincide: a class

*of*

*graphs*defined by a single forbidden induced subgraph H is

*well*-

*quasi*-

*ordered*if and only if it has

*bounded*clique-width if and only if H is an induced ... The conjecture is known to hold for classes

*of*

*graphs*defined by a single forbidden induced subgraph H , as such

*graphs*are

*well*-

*quasi*-

*ordered*and are

*of*

*bounded*clique-width if and only if H is an induced ... One

*of*the first steps towards this result was the proof

*of*the fact that

*graph*classes

*of*

*bounded*

*treewidth*are

*well*-

*quasi*-

*ordered*by the minor relation [26] (a

*graph*parameter π is said to be

*bounded*...

##
###
Well-Quasi-Ordering versus Clique-Width: New Results on Bigenic Classes
[article]

2016
*
arXiv
*
pre-print

Daligault, Rao and Thomassé asked whether a hereditary class

arXiv:1611.03671v1
fatcat:rq6a5tubgbenhhvcluolhthahq
*of**graphs**well*-*quasi*-*ordered*by the induced subgraph relation has*bounded*clique-width. ... The conjecture is known to hold for classes*of**graphs*defined by a single forbidden induced subgraph H, as such*graphs*are*well*-*quasi*-*ordered*and are*of**bounded*clique-width if and only if H is an induced ... One*of*the first steps towards this result was the proof*of*the fact that*graph*classes*of**bounded**treewidth*are*well*-*quasi*-*ordered*by the minor relation [25] (a*graph*parameter π is said to be*bounded*...##
###
Long induced paths in minor-closed graph classes and beyond
[article]

2022
*
arXiv
*
pre-print

*In*this paper we show that every

*graph*

*of*pathwidth less than k that has a path

*of*

*order*n also has an induced path

*of*

*order*at least 1/3 n^1/k. ... This result is then used to prove the two following generalizations: - every

*graph*

*of*

*treewidth*less than k that has a path

*of*

*order*n contains an induced path

*of*

*order*at least 1/4 (log n)^1/k; - for ...

*Well*-

*quasi*-

*ordered*induced subgraphs ideals. Let G be a

*graph*class. We say that G is hereditary if any induced subgraph

*of*a

*graph*

*of*G also belongs to G. ...

##
###
Treewidth versus clique number. I. Graph classes with a forbidden structure
[article]

2021
*
arXiv
*
pre-print

*Treewidth*is an important

*graph*invariant, relevant for both structural and algorithmic reasons. A necessary condition for a

*graph*class to have

*bounded*

*treewidth*is the absence

*of*large cliques. ...

*In*addition, we propose a question about the complexity

*of*the maximum weight independent set problem

*in*(tw,ω)-

*bounded*

*graph*classes and prove that the problem is polynomial-time solvable

*in*every class ... classes, to Jean-Florent Raymond for early discussions on

*treewidth*

*of*1-perfectly orientable

*graphs*, and to Bart M. ...

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