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On the Path Cover Number of Connected Quasi-Claw-Free Graphs

2021
*
IEEE Access
*

INDEX TERMS

doi:10.1109/access.2021.3061600
fatcat:ogoitbwvozdk7f3tp5ykl6o6di
*Path**cover**number*,*quasi*-*claw*-*free**graph*, vertex disjoint*path*. 37086 This work is licensed under a Creative Commons Attribution 4.0 License. ... In this paper, we show that if*the*minimum degree sum*of*an independent set with k + 1 vertices in a*connected**quasi*-*claw*-*free**graph*G*of*order n is no less than n − k, then p(G) ≤ k − 1, where k ≥ 2. ... Clearly,*the*research*on**path**cover**number**of*a*graph*is a generalization*of*determining whether a*graph*is FIGURE 1. ...##
###
Even cycles and perfect matchings in claw-free plane graphs
[article]

2020
*
arXiv
*
pre-print

We show that

arXiv:2001.10674v3
fatcat:qjkkbe7g7ncy3niwqs4pnkffdu
*the*only cycle-nice simple 3-*connected**claw*-*free*plane*graphs*are K_4, W_5 and C_6. ... Furthermore, every cycle-nice 2-*connected**claw*-*free*plane*graph*can be obtained from a*graph*in*the*family F by a sequence*of*three types*of*operations, where F consists*of*even cycles, a diamond, K_4, ... Acknowledgements We would like to thank*the*anonymous referees for their*the*constructive comments and kind suggestions*on*improving*the*representation*of**the*paper. ...##
###
Even cycles and perfect matchings in claw-free plane graphs

2020
*
Discrete Mathematics & Theoretical Computer Science
*

We show that

doi:10.23638/dmtcs-22-4-6
fatcat:tfbq44jknbf5lg5i5oczyamzhm
*the*only cycle-nice simple 3-*connected**claw*-*free*plane*graphs*are $K_4$, $W_5$ and $\overline C_6$. ... Furthermore, every cycle-nice 2-*connected**claw*-*free*plane*graph*can be obtained from a*graph*in*the*family ${\cal F}$ by a sequence*of*three types*of*operations, where ${\cal F}$ consists*of*even cycles ... Acknowledgements We would like to thank*the*anonymous referees for their*the*constructive comments and kind suggestions*on*improving*the*representation*of**the*paper. ...##
###
Page 923 of Mathematical Reviews Vol. , Issue 2000b
[page]

2000
*
Mathematical Reviews
*

*The*author also gives an upper bound for

*the*double competition

*number*

*of*a

*connected*triangle-

*free*

*graph*. ... In this pa- per we determine

*the*K),3-

*free*

*graphs*that are strict

*quasi*-parity and those that are perfectly contractile. ...

##
###
The k-in-a-Path Problem for Claw-free Graphs

2010
*
Algorithmica
*

We show how to solve this problem in polynomial time

doi:10.1007/s00453-010-9468-z
fatcat:76mfyeqqvrfq3jo2ucurnkggmq
*on**claw*-*free**graphs*, when k is not part*of**the*input but an arbitrarily fixed integer. ... Testing whether there is an induced*path*in a*graph*spanning k given vertices is already NP-complete in general*graphs*when k=3. ... Acknowledgments We would like to thank*the*two anonymous referees for their helpful comments. ...##
###
Clique family inequalities for the stable set polytope of quasi-line graphs

2003
*
Discrete Applied Mathematics
*

*Quasi*-line

*graphs*are a subclass

*of*

*claw*-

*free*

*graphs*, and as for

*claw*-

*free*

*graphs*, there exists a polynomial algorithm for finding a maximum weighted stable set

*on*such

*graphs*, but we do not have a complete ... We therefore conjecture that all

*the*non-trivial facets

*of*

*the*ssp

*of*a

*quasi*-line

*graph*belong to this class. ... Acknowledgments:

*The*author is grateful for insightful remarks and encouragement from Antonio Sassano and Bruce Shepherd. ...

##
###
Minimum Weighted Clique Cover on Strip-Composed Perfect Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

*The*only available combinatorial algorithm for

*the*minimum weighted clique

*cover*(mwcc) in

*claw*-

*free*perfect

*graphs*is due to Hsu and Nemhauser [10] and dates back to 1984. ... We also design a new, more efficient, combinatorial algorithm for

*the*mwcc problem

*on*strip-composed

*claw*-

*free*perfect

*graphs*. ... A strip (G, A) is

*claw*-

*free*/

*quasi*-line/line if

*the*

*graph*G + is

*claw*-

*free*/

*quasi*-line/line.

*The*composition

*of*

*claw*-

*free*/

*quasi*-line/line strips is a

*claw*-

*free*/

*quasi*-line/line

*graph*(see e.g. [5] ). ...

##
###
Some Common Properties for Regularizable Graphs, Edge-Critical Graphs and B-Graphs
[chapter]

1982
*
North-Holland Mathematics Studies
*

*The*same proof shows that a

*claw*-

*free*

*graph*with no pendant vertex is

*quasi*-regularizable. So, a

*claw*-

*free*

*graph*with no pendant vertex is

*quasi*-regularizable. ... Sumner [21] have shown independently that a

*connected*

*claw*-

*free*

*graph*with an even

*number*

*of*vertices has a perfect matching (and therefore is

*quasi*-regularizable). ...

##
###
Some common properties for regularizable graphs, edge-critical graphs and b-graphs
[chapter]

1981
*
Lecture Notes in Computer Science
*

*The*same proof shows that a

*claw*-

*free*

*graph*with no pendant vertex is

*quasi*-regularizable. So, a

*claw*-

*free*

*graph*with no pendant vertex is

*quasi*-regularizable. ... Sumner [21] have shown independently that a

*connected*

*claw*-

*free*

*graph*with an even

*number*

*of*vertices has a perfect matching (and therefore is

*quasi*-regularizable). ...

##
###
Finding a smallest odd hole in a claw-free graph using global structure
[article]

2011
*
arXiv
*
pre-print

A lemma

arXiv:1103.6222v2
fatcat:y4oa7m5f2fhuhkqqyknoiaumsi
*of*Fouquet implies that a*claw*-*free**graph*contains an induced C_5, contains no odd hole, or is*quasi*-line. ... In this paper we use this result to give an improved shortest-odd-hole algorithm for*claw*-*free**graphs*by exploiting*the*structural relationship between line*graphs*and*quasi*-line*graphs*suggested by Chudnovsky ... Odd holes are also fundamental to*the*study*of**claw*-*free**graphs*, i.e.*graphs*containing no induced copy*of*K 1,3 . Every neighbourhood v in a*claw*-*free**graph*has stability*number*α(G[N (v)]) ≤ 2. ...##
###
Finding a smallest odd hole in a claw-free graph using global structure

2013
*
Discrete Applied Mathematics
*

A lemma

doi:10.1016/j.dam.2013.04.026
fatcat:omnzksiuvna2db7kftewnjkhlq
*of*Fouquet implies that a*claw*-*free**graph*contains an induced C 5 , contains no odd hole, or is*quasi*-line. ... In this paper we use this result to give an improved shortest-oddhole algorithm for*claw*-*free**graphs*by exploiting*the*structural relationship between line*graphs*and*quasi*-line*graphs*suggested by Chudnovsky ... Odd holes are also fundamental to*the*study*of**claw*-*free**graphs*, i.e.*graphs*containing no induced copy*of*K 1,3 . Every neighbourhood v in a*claw*-*free**graph*has stability*number*α(G[N (v)]) ≤ 2. ...##
###
An algorithmic decomposition of claw-free graphs leading to anO(n3)-algorithm for the weighted stable set problem
[chapter]

2011
*
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms
*

We also believe that our algorithmic decomposition result is interesting

doi:10.1137/1.9781611973082.49
dblp:conf/soda/FaenzaOS11
fatcat:jkcj6m5rhjextbqoxsdwvv453a
*on*its own and might be also useful to solve other kind*of*problems*on**claw*-*free**graphs*. ... This algorithm is based*on*a novel decomposition theorem for*claw*-*free**graphs*, which is also introduced in*the*present paper. ... Let 0 be a*connected**claw*-*free*but not*quasi*-line*graph*with vertices. ...##
###
Hamilton connectedness and the partially square graphs

2006
*
Discrete Mathematics
*

A -

doi:10.1016/j.disc.2005.11.021
fatcat:4cnozrtagjchffn5udxadbk5na
*connected*( 3)*quasi*-*claw*-*free*(or a*claw*-*free*)*graph*G*of*order n is hamiltonian-*connected*if (G 2 ) < . Corollary 1.6. ... A -*connected*( 3)*quasi*-*claw*-*free*(or a*claw*-*free*)*graph*G*of*order n is hamiltonian-*connected*if S > n − holds for every S ∈ I +1 (G). Theorem 1.7. ... Acknowledgements*The*authors like to thank*the*anonymous referees for their helpful and pertinent suggestions. ...##
###
Page 6363 of Mathematical Reviews Vol. , Issue 2002I
[page]

2002
*
Mathematical Reviews
*

Atsushi; Kawarabayashi, Ken-ichi (J-K EIOE; Yokohama);
Matsuda, Haruhide

*Path*factors in*claw*-*free**graphs*. ... We prove that if G is a*claw*-*free**graph*with minimum degree at least d, then G has a*path*factor g(u) > 1 for at least half*of*2002i:05095 such that*the*order*of*each*path*is at least d +1.” 2002i1:05091 ...##
###
Colouring Generalized Claw-Free Graphs and Graphs of Large Girth: Bounding the Diameter
[article]

2021
*
arXiv
*
pre-print

A

arXiv:2111.11897v1
fatcat:w76s7sr3oze47fuxugd5767g4i
*graph*G is H-*free*if G does not contain H as an induced subgraph. It is known that for all k≥ 3,*the*k-Colouring problem is NP-complete for H-*free**graphs*if H contains an induced*claw*or cycle. ...*The*case where H contains a cycle follows from*the*known result that*the*problem is NP-complete even for*graphs**of*arbitrarily large fixed girth. ... Note that*claw*-*free**graphs*are locally*claw*-*free*,*quasi*-*claw*-*free*and almost*claw*-*free*. Hence,*the*latter three*graph*classes all generalize*the*class*of**claw*-*free**graphs*. ...
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