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Connected k-partition of k-connected graphs and c-claw-free graphs
[article]

2021
*
arXiv
*
pre-print

We study BCP on

arXiv:2107.04837v1
fatcat:c3vig6cu3jasdfu5waklikpwfi
*c*-*claw*-*free**graphs*, the class*of**graphs*that do not have K_1,*c*as an induced subgraph,*and*present efficient (*c*-1)-approximation algorithms for both objectives. ... A*connected**partition*is a*partition**of*the vertices*of*a*graph*into sets that induce*connected*subgraphs. ... ▶ Lemma 9 . 9 Given a*K*1,*c*-*free**graph*G*and*a DFS-tree*of*G. ...##
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Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

2021

We study BCP on

doi:10.4230/lipics.approx/random.2021.27
fatcat:44xgybpmfvbu7gaxht3n5tdmnu
*c*-*claw*-*free**graphs*, the class*of**graphs*that do not have K_{1,*c*} as an induced subgraph,*and*present efficient (*c*-1)-approximation algorithms for both objectives. ... In particular, for 3-*claw*-*free**graphs*, also simply known as*claw*-*free**graphs*, we obtain a 2-approximation. ... We say a*graph*is*c*-*claw*-*free*or*K*1,*c*-*free*if it does not contain an induced*K*1,*c*as subgraph. ...##
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The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

2018
*
Graphs and Combinatorics
*

Moreover, we also prove that

doi:10.1007/s00373-018-1928-2
fatcat:errn36spirhsvisjty3hk7tel4
*claw*-*free*cographs, that is, {K_1,3,P_4} -*free**graphs*belong to a known class*of*e -positive*graphs*. ... Stanley defined the chromatic symmetric function*of*a simple*graph**and*has conjectured that every tree is determined by its chromatic symmetric function. ... Every coconnected component*of*a*connected**claw*-*free*cograph is*K*1 or a disjoint union*of*two complete*graphs*. Proof. Let G be a*connected**claw*-*free**graph*. ...##
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2-connected claw-free chordal graphs are cycle extendable
[article]

2016
*
arXiv
*
pre-print

We prove that any 2-

arXiv:1310.2901v4
fatcat:65rhyhjiafgmnpb7pozb52zkeq
*connected**claw*-*free*chordal*graph*is cycle extendable. ... A cycle*C**of*length*k*in*graph*G is extendable if there is another cycle*C*' in G with V(*C*) ⊂ V(*C*')*and*length*k*+1. A*graph*is cycle extendable if every non-Hamiltonian cycle is extendable. ...*And*also a result*of*Oberly*and*Sumner [10] : Lemma 2.2 Any 2-*connected*locally*connected**graph*which is*claw*-*free*is Hamiltonian. ...##
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Recognizing Polar Planar Graphs Using New Results for Monopolarity
[chapter]

2011
*
Lecture Notes in Computer Science
*

A

doi:10.1007/978-3-642-25591-5_14
fatcat:7qs2oj3afjacnmg7p2d7avkkia
*graph*G = (V, E) is polar if its vertex set admits a*partition*V = A∪B such that A induces a complete multipartite*and*B the complement*of*a complete multipartite*graph*. ... Polar*and*monopolar*graphs*are natural generalizations*of*bipartite or split*graphs*. ... A path with*k*vertices*and**k*− 1 edges is denoted by P*k*, a cycle*of*length*k*denoted by*C**k*. A*C*3 is also called a triangle*and*a*C**k*,*k*≥ 5, is also called a hole. ...##
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On the complexity of the dominating induced matching problem in hereditary classes of graphs

2011
*
Discrete Applied Mathematics
*

In particular, we develop polynomial-time algorithms to solve the problem for

doi:10.1016/j.dam.2010.03.011
fatcat:q236falzp5e3rmqa3bxaolpddy
*claw*-*free**graphs**and*convex*graphs*. ... ., a class lying on a "boundary" separating difficult instances*of*the problem from polynomially solvable ones)*and*derive a number*of*polynomial-time results. ... Cardoso was supported by the Centre*of*Research on Optimization*and*Control from the Fundação para a Ciência e Tecnologia cofinanced by the European Community Fund FEDER/POCI/2010*and*the research*of*Vadim ...##
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EVERY N2-LOCALLY CONNECTED CLAW-FREE GRAPH WITH MINIMUM DEGREE AT LEAST 7 IS Z3-CONNECTED

2011
*
Discrete Mathematics, Algorithms and Applications (DMAA)
*

Each

doi:10.1142/s1793830911001140
fatcat:3qdv6ulhc5ekxlspbku2vgz3qa
*K*n in Fig. 2 represents 7 a complete*graph*with n ≥ 6. Then G is a*claw*-*free**graph*with δ(G) ≥ 7,*and*8 G is not N 2 -locally*connected*. By Lemma 2.2,*K*n*and**C*2 is Z 3 -*connected*. ... If E(H*c*) = ∅, since G is a*claw*-*free**graph*,19 every path in H*c*has length at most 1. Thus H*c*is the union*of*disjoint edges (*and*20 some isolated vertices). ...##
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Triangulation and Clique Separator Decomposition of Claw-Free Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

We propose a recursive algorithm which works for general

doi:10.1007/978-3-642-34611-8_5
fatcat:fzsaq3knunc5vfahlr6dnijsii
*graphs**and*runs in linear time if the input is a*claw*-*free**graph**and*the length*of*its longest path is bounded by a fixed value*k*. ... More precisely, our algorithm runs in O(f + km) time if the input is a*claw*-*free**graph*, where f is the number*of*fill edges added,*and**k*is the height*of*the execution tree; we find all the clique minimal ... In a*claw*-*free**graph*G = (V, E), any vertex x belonging to a moplex defines only one substar S. Moreover,*C*= V − N [x] induces a*connected**graph*,*and*S = N (*C*). ...##
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Two results on matching extensions with prescribed and proscribed edge sets

1999
*
Discrete Mathematics
*

The ÿrst involves the class

doi:10.1016/s0012-365x(98)00405-1
fatcat:wrf3ctslk5hrtfaikpthdtnoiq
*of**claw*-*free**graphs**and*the second is a result about bipartite*graphs*. . 0012-365X/99/$ -see front matter*c*1999 Elsevier Science B.V. ... Then G is E(m; n) if for each pair*of*disjoint matchings M; N ⊆ E(G)*of*size m*and*n, respectively, there exists a perfect matching F in G such that M ⊆ F*and*F ∩ N = ∅. ...*Claw*-*free**graphs*have been widely studied in*connection*with many*graph*parameters*and*concepts, only one*of*which is matching. ...##
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Resolving Stanley's e-positivity of claw-contractible-free graphs
[article]

2020
*
arXiv
*
pre-print

We resolve this by giving infinite families

arXiv:1703.05770v3
fatcat:uu2gf2bxtventdwka5epyo6nce
*of**graphs*that are not contractible to the*claw**and*whose chromatic symmetric functions are not e-positive. ... Moreover, one such family is additionally*claw*-*free*, thus establishing that the e-positivity*of*chromatic symmetric functions is in general not dependent on the existence*of*an induced*claw*or*of*a contraction ... All three authors were supported in part by the National Sciences*and*Engineering Research Council*of*Canada. This work was supported by the Canadian Tri-Council Research Support Fund. ...##
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A revision and extension of results on 4-regular, 4-connected, claw-free graphs

2011
*
Discrete Applied Mathematics
*

In this paper, we will complete Plummer's characterization

doi:10.1016/j.dam.2011.04.013
fatcat:pnq4z4lglrcz3djonrq6yaqsly
*of*all 4-*connected*, 4-regular,*claw*-*free**graphs*,*and*then show the implications this has on the well-covered*graphs*he*and*Hartnell determined ... In 1995 , Plummer (1992 [6] published a paper in which he gave a characterization*of*4regular, 4-*connected*,*claw*-*free**graphs*. ... If G is 4-*connected*, 4-regular,*claw*-*free**and*contains a*K*4 , then either G =*K*5 or the vertex set V (G) can be*partitioned*into disjoint sets*of*four vertices each, such that each four-vertex set induces ...##
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Locating-total domination in claw-free cubic graphs

2012
*
Discrete Mathematics
*

We show that the locating-total domination number

doi:10.1016/j.disc.2012.06.024
fatcat:vtan26p4wne2ljpk4anow2hcua
*of*a*claw*-*free*cubic*graph*is at most one-half its order*and*we characterize the*graphs*achieving this bound. ... A clawfree*graph*is a*graph*that does not contain*K*1,3 as an induced subgraph. ... Acknowledgments The research*of*the first author was supported in part by the South African National Research Foundation*and*the University*of*Johannesburg,*and*the research*of*the second author was supported ...##
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A note on contracting claw-free graphs

2013
*
Discrete Mathematics & Theoretical Computer Science
*

We show that these problems stay NP-complete even when the host

doi:10.46298/dmtcs.605
fatcat:hrmcp2vu7jehnm432miqaeucam
*and*target belong to the class*of*line*graphs*, which form a subclass*of*the class*of**claw*-*free**graphs*, i.e.,*graphs*with no induced 4-vertex ... We show that these problems may become computationally easier when the host*graphs*are restricted to be*claw*-*free*. ... Acknowledgements We thank both anonymous reviewers for their useful comments that helped us to improve the readability*of*our paper. ...##
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Zero Forcing in Claw-Free Cubic Graphs
[article]

2018
*
arXiv
*
pre-print

As a consequence

arXiv:1802.03108v1
fatcat:vyrdfeghtrecbftggmnmelopfu
*of*this result, if G K_4 is a*connected*, cubic,*claw*-*free**graph*with order n, then Z(G) <2/5n + 1. ... Our main theorem states that if G K_4 is a*connected*, cubic,*claw*-*free**graph*, then Z(G) <α(G) + 1. This improves on best known upper bounds for Z(G), as well as known lower bounds on α(G). ... Figure 1 : The prism*C*3 ✷*K*2*and*the diamond-necklaces N 2*and*N 3 . (a)*C*3 ✷*K*2 (b) N 2 (*c*) N 3 The following property*of**connected*,*claw*-*free*, cubic*graphs*is established in [20] . ...##
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Disconnected Cuts in Claw-free Graphs

2018
*
European Symposium on Algorithms
*

Its

doi:10.4230/lipics.esa.2018.61
dblp:conf/esa/MartinPL18
fatcat:znrxo6i2v5abxlkgdmb5pnrznu
*connection*to the complexity*of*the problem to contract a*claw*-*free**graph*to the 4-vertex cycle*C*4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. ... However, the complexity*of*the problem on*claw*-*free**graphs*remained an open question. ... Note that every line*graph*is*claw*-*free*. We call G the preimage*of*L(G). Every*connected*line*graph*except*K*3 has a unique preimage [17] . ...
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