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

Huiqing Liu, Jian Lu, Shunzhe Zhang, Xiuyu Zhong
2021 IEEE Access  
INDEX TERMS 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.  ... 
doi:10.1109/access.2021.3061600 fatcat:ogoitbwvozdk7f3tp5ykl6o6di

Even cycles and perfect matchings in claw-free plane graphs [article]

Shanshan Zhang, Xiumei Wang, Jinjiang Yuan
2020 arXiv   pre-print
We show that 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.  ... 
arXiv:2001.10674v3 fatcat:qjkkbe7g7ncy3niwqs4pnkffdu

Even cycles and perfect matchings in claw-free plane graphs

Shanshan Zhang, Xiumei Wang, Jinjiang Yuan
2020 Discrete Mathematics & Theoretical Computer Science  
We show that 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.  ... 
doi:10.23638/dmtcs-22-4-6 fatcat:tfbq44jknbf5lg5i5oczyamzhm

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

Jiří Fiala, Marcin Kamiński, Bernard Lidický, Daniël Paulusma
2010 Algorithmica  
We show how to solve this problem in polynomial time 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.  ... 
doi:10.1007/s00453-010-9468-z fatcat:76mfyeqqvrfq3jo2ucurnkggmq

Clique family inequalities for the stable set polytope of quasi-line graphs

G. Oriolo
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.  ... 
doi:10.1016/s0166-218x(03)00400-1 fatcat:vai3up33sngjtot2itdwfyzt5y

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

Flavia Bonomo, Gianpaolo Oriolo, Claudia Snels
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] ).  ... 
doi:10.1007/978-3-642-34611-8_6 fatcat:bhtkybrokvbsvb2fkiw7qk55qy

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

Claude Berge
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).  ... 
doi:10.1016/s0304-0208(08)73488-1 fatcat:pnpql5vfz5cu3icliw6ogxo7wy

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

Claude Berge
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).  ... 
doi:10.1007/3-540-10704-5_10 fatcat:zjnrx6kgu5b6dlqqwioylx5yby

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

W. Sean Kennedy, Andrew D. King
2011 arXiv   pre-print
A lemma 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.  ... 
arXiv:1103.6222v2 fatcat:y4oa7m5f2fhuhkqqyknoiaumsi

Finding a smallest odd hole in a claw-free graph using global structure

Wm. Sean Kennedy, Andrew D. King
2013 Discrete Applied Mathematics  
A lemma 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.  ... 
doi:10.1016/j.dam.2013.04.026 fatcat:omnzksiuvna2db7kftewnjkhlq

An algorithmic decomposition of claw-free graphs leading to anO(n3)-algorithm for the weighted stable set problem [chapter]

Yuri Faenza, Gianpaolo Oriolo, Gautier Stauffer
2011 Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms  
We also believe that our algorithmic decomposition result is interesting 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.  ... 
doi:10.1137/1.9781611973082.49 dblp:conf/soda/FaenzaOS11 fatcat:jkcj6m5rhjextbqoxsdwvv453a

Hamilton connectedness and the partially square graphs

Ahmed Ainouche, Serge Lapiquonne
2006 Discrete Mathematics  
A -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.  ... 
doi:10.1016/j.disc.2005.11.021 fatcat:4cnozrtagjchffn5udxadbk5na

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]

Barnaby Martin and Daniel Paulusma and Siani Smith
2021 arXiv   pre-print
A 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.  ... 
arXiv:2111.11897v1 fatcat:w76s7sr3oze47fuxugd5767g4i
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