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

Ralf Borndörfer, Katrin Casel, Davis Issac, Aikaterini Niklanovits, Stephan Schwartz, Ziena Zeif
2021 arXiv   pre-print
We study BCP on 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.  ... 
arXiv:2107.04837v1 fatcat:c3vig6cu3jasdfu5waklikpwfi

Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

Ralf Borndörfer, Katrin Casel, Davis Issac, Aikaterini Niklanovits, Stephan Schwartz, Ziena Zeif, Mary Wootters, Laura Sanità
We study BCP on 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.  ... 
doi:10.4230/lipics.approx/random.2021.27 fatcat:44xgybpmfvbu7gaxht3n5tdmnu

The Chromatic Symmetric Functions of Trivially Perfect Graphs and Cographs

Shuhei Tsujie
2018 Graphs and Combinatorics  
Moreover, we also prove that 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.  ... 
doi:10.1007/s00373-018-1928-2 fatcat:errn36spirhsvisjty3hk7tel4

2-connected claw-free chordal graphs are cycle extendable [article]

Deborah Arangno, David E. Brown
2016 arXiv   pre-print
We prove that any 2-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.  ... 
arXiv:1310.2901v4 fatcat:65rhyhjiafgmnpb7pozb52zkeq

Recognizing Polar Planar Graphs Using New Results for Monopolarity [chapter]

Van Bang Le, Ragnar Nevries
2011 Lecture Notes in Computer Science  
A 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.  ... 
doi:10.1007/978-3-642-25591-5_14 fatcat:7qs2oj3afjacnmg7p2d7avkkia

On the complexity of the dominating induced matching problem in hereditary classes of graphs

Domingos M. Cardoso, Nicholas Korpelainen, Vadim V. Lozin
2011 Discrete Applied Mathematics  
In particular, we develop polynomial-time algorithms to solve the problem for 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  ... 
doi:10.1016/j.dam.2010.03.011 fatcat:q236falzp5e3rmqa3bxaolpddy


2011 Discrete Mathematics, Algorithms and Applications (DMAA)  
Each 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).  ... 
doi:10.1142/s1793830911001140 fatcat:3qdv6ulhc5ekxlspbku2vgz3qa

Triangulation and Clique Separator Decomposition of Claw-Free Graphs [chapter]

Anne Berry, Annegret Wagler
2012 Lecture Notes in Computer Science  
We propose a recursive algorithm which works for general 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).  ... 
doi:10.1007/978-3-642-34611-8_5 fatcat:fzsaq3knunc5vfahlr6dnijsii

Two results on matching extensions with prescribed and proscribed edge sets

R.E.L. Aldred, D.A. Holton, M.I. Porteous, Michael D. Plummer
1999 Discrete Mathematics  
The ÿrst involves the class 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.  ... 
doi:10.1016/s0012-365x(98)00405-1 fatcat:wrf3ctslk5hrtfaikpthdtnoiq

Resolving Stanley's e-positivity of claw-contractible-free graphs [article]

Samantha Dahlberg, Angele Foley, Stephanie van Willigenburg
2020 arXiv   pre-print
We resolve this by giving infinite families 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.  ... 
arXiv:1703.05770v3 fatcat:uu2gf2bxtventdwka5epyo6nce

A revision and extension of results on 4-regular, 4-connected, claw-free graphs

Trevor J. Gionet, Erika L.C. King, Yixiao Sha
2011 Discrete Applied Mathematics  
In this paper, we will complete Plummer's characterization 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  ... 
doi:10.1016/j.dam.2011.04.013 fatcat:pnq4z4lglrcz3djonrq6yaqsly

Locating-total domination in claw-free cubic graphs

Michael A. Henning, Christian Löwenstein
2012 Discrete Mathematics  
We show that the locating-total domination number 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  ... 
doi:10.1016/j.disc.2012.06.024 fatcat:vtan26p4wne2ljpk4anow2hcua

A note on contracting claw-free graphs

Jiří Fiala, Marcin Kamiński, Daniël Paulusma
2013 Discrete Mathematics & Theoretical Computer Science  
We show that these problems stay NP-complete even when the host 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.  ... 
doi:10.46298/dmtcs.605 fatcat:hrmcp2vu7jehnm432miqaeucam

Zero Forcing in Claw-Free Cubic Graphs [article]

Randy Davila, Michael Henning
2018 arXiv   pre-print
As a consequence 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] .  ... 
arXiv:1802.03108v1 fatcat:vyrdfeghtrecbftggmnmelopfu

Disconnected Cuts in Claw-free Graphs

Barnaby Martin, Daniël Paulusma, Erik Jan Van Leeuwen, Michael Wagner
2018 European Symposium on Algorithms  
Its 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] .  ... 
doi:10.4230/lipics.esa.2018.61 dblp:conf/esa/MartinPL18 fatcat:znrxo6i2v5abxlkgdmb5pnrznu
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