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An Efficient Algorithm for Enumerating Chordal Bipartite Induced Subgraphs in Sparse Graphs
[article]

2019
*
arXiv
*
pre-print

*In*this paper, we propose a characterization of

*chordal*

*bipartite*

*graphs*and

*an*

*efficient*

*enumeration*

*algorithm*

*for*

*chordal*

*bipartite*

*induced*

*subgraphs*. ... Moreover, we propose

*an*

*algorithm*ECB which

*enumerates*all

*chordal*

*bipartite*

*induced*

*subgraphs*

*in*O(ktΔ^2) time per solution on average, where k is the degeneracy, t is the maximum size of K_t,t as

*an*... We consider the number of edges

*in*B[N 1: , the sum of |N (u i )| is at most O(t∆). We next consider the case

*for*1 ≤ i ≤ d(v) − t. ...

##
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Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs
[article]

2021
*
arXiv
*
pre-print

We combine direct proof techniques with our

arXiv:2107.12031v1
fatcat:s4waxe2m7be73cnphhiigliyfy
*efficient**graph*generation*algorithm*to compute several new defective Ramsey numbers*in*perfect*graphs*,*bipartite**graphs*and*chordal**graphs*. ... We obtain several values*for*c^𝒢_k(m)*in*perfect*graphs*and cographs. ... We run*Algorithm*1 with these settings and observe that there is no*bipartite**graph*on 18 vertices that has*an**induced**subgraph*from T BIP m (1, 4, 9)*for*m ≥ 14. ...##
###
Recent developments on graphs of bounded clique-width

2009
*
Discrete Applied Mathematics
*

Whether the clique-width of

doi:10.1016/j.dam.2008.08.022
fatcat:h45w5c3pfrazbgyfboniyohhha
*graphs**in*a certain class of*graphs*is bounded or not, is*an*important question from*an**algorithmic*point of view, as many problems that are NPhard*in*general admit polynomial-time ... Over the last few years, many classes of*graphs*have been shown to have bounded clique-width.*For*many others, this parameter has been proved to be unbounded. ... This nice decomposability property has led to*efficient*solutions*for*many*algorithmic**graph*problems that are NP-hard*in*general [13] . ...##
###
Graph classes and forbidden patterns on three vertices
[article]

2020
*
arXiv
*
pre-print

Unfortunately this strategy usually does not lead to

arXiv:1812.05913v3
fatcat:nmuipxm33fbifkjraluex2i3sm
*an**efficient*recognition*algorithm*. ... This paper deals with*graph*classes characterization and recognition. A popular way to characterize a*graph*class is to list a minimal set of forbidden*induced**subgraphs*. ... The first author is thankful to José Correa, Flavio Guinez and Mauricio Soto,*for*the very first discussions on this topic back*in*2013. ...##
###
Page 3893 of Mathematical Reviews Vol. , Issue 97F
[page]

1997
*
Mathematical Reviews
*

The results include

*efficient*parallel*algorithms**for*recognizing*chordal**graphs*, finding a maximum clique and a maximum independent set*in*a*chordal**graph*, finding a breadth-first search tree and a depth ... We also show*an*upper bound of roughly | + Vk —1 on the average degree of linear-sized*induced**subgraphs*of Ramanujan*graphs*. This compares positively with the classi- cal bound 2k — 1. ...##
###
Large Induced Subgraphs via Triangulations and CMSO

2015
*
SIAM journal on computing (Print)
*

*For*a given

*graph*G = (V, E), the task is to maximize |X| subject to the following: there is a set F ⊆ V such that X ⊆ F , the

*subgraph*G[F ]

*induced*by F is of treewidth at most t, and structure (G[F ... We obtain

*an*

*algorithmic*meta-theorem

*for*the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t ≥ 0 be

*an*integer. ... Thilikos

*for*fruitful discussions and useful suggestions on the topic of the paper. ...

##
###
Treewidth versus clique number. II. Tree-independence number
[article]

2022
*
arXiv
*
pre-print

*algorithmic*implications

*for*problems related to independent sets. ... We define the independence number of a tree decomposition 𝒯 of a

*graph*as the maximum independence number over all

*subgraphs*of G

*induced*by some bag of 𝒯. ... Acknowledgements This work is supported

*in*part by the Slovenian Research Agency (I0-0035, research programs P1-0285 and P1-0383, research projects J1-3001, J1-3002, J1-3003, J1-9110, N1-0102, N1-0160, ...

##
###
Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph
[article]

2006
*
arXiv
*
pre-print

We characterize clique trees of a

arXiv:cs/0607055v1
fatcat:5phyhwym6rbrpez7f35gb74p6u
*chordal**graph**in*their relation to simplicial vertices and perfect sequences of maximal cliques. ... We describe the relation as a*bipartite**graph*and prove that the*bipartite**graph*is always connected. ... This fact is important*in**enumerating*all clique trees*for*a given*chordal**graph*. ...##
###
Large induced subgraphs via triangulations and CMSO
[article]

2013
*
arXiv
*
pre-print

We give

arXiv:1309.1559v1
fatcat:f6ljhfu3dbdtdf32yr7ljjf4pe
*an**algorithm*solving the optimization problem on*an*n-vertex*graph*G*in*time O(#pmc n^t+4 f(t,ϕ)), where #pmc is the number of all potential maximal cliques*in*G and f is a function depending of ... We obtain*an**algorithmic*meta-theorem*for*the following optimization problem. Let ϕ be a Counting Monadic Second Order Logic (CMSO) formula and t be*an*integer. ... Thilikos*for*fruitful discussions and useful suggestions on the topic of the paper. ...##
###
An efficient parallel algorithm for the minimal elimination ordering (MEO) of an arbitrary graph

1994
*
Theoretical Computer Science
*

We design the first

doi:10.1016/0304-3975(94)90250-x
fatcat:u5whyapbyvdepc4xzo4iszbdxm
*efficient*parallel*algorithm**for*computing the minimal elimination ordering (MEO) of*an*arbitrary*graph*. ... Karpinski,*An**efficient*parallel*algorithm**for*the minimal elimination ordering (MEO) of*an*arbitrary*graph*, Theoretical Computer Science 134 (1994) 493-528. ... Acknowledgements Bob Tarjan has raised to us the question of*an**efficient*parallel*algorithm**for*the ME0 Problem. We are thankful to him, and also to Philip Klein*for*the stimulating discussions. ...##
###
Vertex Deletion Parameterized by Elimination Distance and Even Less
[article]

2022
*
arXiv
*
pre-print

They are related to

arXiv:2103.09715v4
fatcat:vv3kluwdtjhqdhaflntr6msvdi
*graph*decompositions*in*which*subgraphs*that belong to a target class H (e.g.,*bipartite*or planar) are considered simple. ... This is applicable to any*graph*class H*for*which the corresponding vertex-deletion problem admits a constant-factor approximation*algorithm*or*an*FPT*algorithm*paramaterized by the solution size. ...*For*H defined by a finite set of forbidden*induced**subgraphs*such that Vertex cover is polynomial-time solvable on*graphs*from H (*for*example, claw-free*graphs*), or H ∈ {*chordal*,*bipartite*} (*for*which ...##
###
New Applications of Clique Separator Decomposition for the Maximum Weight Stable Set Problem
[chapter]

2005
*
Lecture Notes in Computer Science
*

*Graph*decompositions such as decomposition by clique separators and modular decomposition are of crucial importance

*for*designing

*efficient*

*graph*

*algorithms*. ...

*In*particular, we describe the atom structure

*for*some new classes of

*graphs*whose atoms are P5-free (the P5 is the

*induced*path with 5 vertices) and obtain new polynomial time results

*for*MWS. ...

*For*

*an*

*induced*

*subgraph*H of G, a vertex not

*in*H is a k-vertex of H, if it has exactly k neighbors

*in*H. A

*graph*is

*chordal*if it contains no

*induced*cycle C k , k ≥ 4. ...

##
###
Counting the number of independent sets in chordal graphs

2008
*
Journal of Discrete Algorithms
*

We first provide the following

doi:10.1016/j.jda.2006.07.006
fatcat:pt3o77gc5zcrhdirjtbnrx5kwe
*efficient**algorithms**for*a*chordal**graph*: (1) a linear-time*algorithm**for*counting the number of independent sets; (2) a linear-time*algorithm**for*counting the number of ... We study some counting and*enumeration*problems*for**chordal**graphs*, especially concerning independent sets. ... Acknowledgement The authors thank Masashi Kiyomi*for*enlightening discussions and pointing out the work by Chang [7] . The authors are grateful to L. Shankar Ram*for*pointing out a paper [5] . ...##
###
Linear-Time Counting Algorithms for Independent Sets in Chordal Graphs
[chapter]

2005
*
Lecture Notes in Computer Science
*

We first provide the following

doi:10.1007/11604686_38
fatcat:dvnk575gnzgyrdcnzombvtlojm
*efficient**algorithms**for*a*chordal**graph*: (1) a linear-time*algorithm**for*counting the number of independent sets; (2) a linear-time*algorithm**for*counting the number of ... We study some counting and*enumeration*problems*for**chordal**graphs*, especially concerning independent sets. ... Shankar Ram*for*pointing out a paper [5] . References ...##
###
Hardness Results and an Exact Exponential Algorithm for the Spanning Tree Congestion Problem
[chapter]

2011
*
Lecture Notes in Computer Science
*

Additionally, we provide a constant-factor approximation

doi:10.1007/978-3-642-20877-5_44
fatcat:nim2cohsdjewrmawau43sjoysy
*algorithm**for*cographs, and a linear-time*algorithm**for**chordal*cographs. ⋆ ... This paper studies the complexity of the problem to determine the spanning tree congestion*for*non-*sparse**graph*classes, while it was investigated*for*some*sparse**graph*classes before. ... Remarks on cographs We showed NP-completeness of STC*for**graphs*of clique-width at most three. ...
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