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Disjoint Paired-Dominating sets in Cubic Graphs

Gábor Bacsó, Csilla Bujtás, Casey Tompkins, Zsolt Tuza
2019 Graphs and Combinatorics  
We prove that the vertex set of every claw-free cubic graph can be partitioned into two paired-dominating sets.  ...  A paired-dominating set of a graph G is a dominating set D with the additional requirement that the induced subgraph G[D] contains a perfect matching.  ...  The graph invariant asked for in Problem 3(1) is always positive because every nonextendable matching induces a paired-dominating set whenever isolates are excluded.  ... 
doi:10.1007/s00373-019-02063-w pmid:31631942 pmcid:PMC6777509 fatcat:mfnujdt5dvcadhcysxyxp4tbam

How Many Conjectures Can You Stand? A Survey

H. J. Broersma, Z. Ryjáček, P. Vrána
2011 Graphs and Combinatorics  
We survey results and open problems in hamiltonian graph theory centered around two conjectures of the 1980s that are still open: every 4-connected claw-free graph (line graph) is hamiltonian.  ...  of claw-free graphs.  ...  line (claw-free) graphs.  ... 
doi:10.1007/s00373-011-1090-6 fatcat:jj7oxvyx5vhrzdjtv2qyj7nysy

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.  ...  In this paper, we continue the study of locating-total domination in graphs. A set S of vertices of a graph G is a total dominating set of G if every vertex of G is adjacent to a vertex in S.  ...  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

On some intriguing problems in hamiltonian graph theory—a survey

H.J. Broersma
2002 Discrete Mathematics  
We survey results and open problems in hamiltonian graph theory centered around three themes: regular graphs, t-tough graphs, and claw-free graphs.  ...  The arguments used to prove the equivalence of Conjectures 4.2 and 4.22 can be combined with the claw-free closure operation to obtain results on cycles in claw-free graphs from results on cycles in cubic  ...  In the sequel we will focus on cyclic and other properties of cubic (i.e., 3-regular) graphs, and show their close relationship with results and conjectures on line graphs and claw-free graphs.  ... 
doi:10.1016/s0012-365x(01)00325-9 fatcat:wcggfjuntbgrjmegqldzdz55hy

Boxicity and cubicity of asteroidal triple free graphs

Diptendu Bhowmick, L. Sunil Chandran
2010 Discrete Mathematics  
Upper bound on boxicity of AT-free graphs and cubicity of claw-free AT-free graphs In this section, we obtain upper bounds on boxicity of AT-free graphs and cubicity of claw-free AT-free graphs.  ...  A pair of vertices x, y is a dominating pair if all x, y-paths in G are dominating sets.  ... 
doi:10.1016/j.disc.2010.01.020 fatcat:blgsakh46jdsrodwtnp4ugcaq4

Packing 3-vertex paths in claw-free graphs and related topics

Alexander Kelmans
2011 Discrete Applied Mathematics  
Keywords: claw-free graph, cubic graph, vertex disjoint packing, edge disjoint packing, 3-vertex factor, 3-vertex packing, path-factor, induced packing, graph domination, graph minor, the Hadwiger conjecture  ...  Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G.  ...  There are infinitely many pairs (G, L) such that G is a 3-connected, claw-free, and non-cubic graph, v(G) ≡ 0 mod 3, L is a 3-vertex path in G, and G − L has no -factor.  ... 
doi:10.1016/j.dam.2010.05.001 fatcat:74tl7ztnxrdt3ccymbrplkpbay

Boxicity and Cubicity of Asteroidal Triple free graphs [article]

Diptendu Bhowmick, L. Sunil Chandran
2009 arXiv   pre-print
Let G be an AT-free graph with chromatic number χ(G) and claw number . In this paper we will show that (G) ≤χ(G) and this bound is tight. We also show that (G) ≤(G)(_2 +2) ≤ χ(G)(_2 +2).  ...  If G is an AT-free graph having girth at least 5 then (G) ≤ 2 and therefore (G) ≤ 2_2 +4.  ...  on boxicity of AT-free graphs and cubicity of claw-free AT-free graphs.  ... 
arXiv:0812.0894v2 fatcat:mnrbwranvbayhl3eq652xbffqi

On a conjecture on total domination in claw-free cubic graphs

Justin Southey, Michael A. Henning
2010 Discrete Mathematics  
Cockayne et al. [9] showed that every claw-free cubic graph has total domination number one-half its order. Theorem 2 ([9]). If G is a claw-free cubic graph of order n, then γ t (G) ≤ n/2.  ...  The connected claw-free cubic graphs that achieve equality in the bound of Theorem 2 are characterized in [10] .  ...  Acknowledgement The research of both authors was supported in part by the South African National Research Foundation.  ... 
doi:10.1016/j.disc.2010.07.006 fatcat:w5wsqvc3avhk3hfrvds3d7rfum

Packing 3-Vertex Paths in Claw-Free Graphs [article]

Alexander Kelmans
2007 arXiv   pre-print
Keywords: claw-free graph, cubic graph, L-packing, L-factor.  ...  Let G be a 3-connected claw-free graph, x be a vertex, e = xy be an edge, and P be a 3-vertex path in G.  ...  There are infinitely many pairs (G, L) such that G is a 3-connected, claw-free, and non-cubic graph, v(G) = 0 mod 3, L is a 3-vertex path in G, and G − L has no Λ-factor.  ... 
arXiv:0711.3871v1 fatcat:gcyai76ty5ci7p3yqyr7jja2hm

Bounds on total domination in claw-free cubic graphs

Odile Favaron, Michael A. Henning
2008 Discrete Mathematics  
However, every graph in these two families, except for K 4 and a cubic graph of order eight, contains a claw.  ...  In this paper, we answer this question in the affirmative. We prove that if G is a connected claw-free cubic graph of order n 10, then t (G) 5n/11. (O. Favaron), henning@ukzn.ac.za (M.A. Henning).  ...  Final remark (concerning paired domination): In a previous paper [5] we proved that if a connected claw-free cubic graph of order n 6 does not contain K 4 − e nor C 4 as an induced subgraph, then its  ... 
doi:10.1016/j.disc.2007.07.007 fatcat:3ayowy544veadgrxyxlfekmvsu

Classes of cubic graphs containing cycles of integer-power lengths

P. J. Couch, Bobby Dale Daniel, W. Paul Wright
2021 The Australasian Journal of Combinatorics  
Erdős and Gyárfás conjectured in 1995 that every graph with minimum degree three has a cycle of length 2 k for some integer k > 1. Y.  ...  We address a special case of the question of Caro, as well as others, by showing that every graph G of minimum degree 3, such that the set of centers of induced claws of G is independent, contains a cycle  ...  It is straightforward to see that every claw-free graph is almost claw-free. Let G be an almost claw-free cubic graph.  ... 
dblp:journals/ajc/CouchDW21 fatcat:2zq7a6qczff6pjk5ddvd4gobm4

Some recent results on domination in graphs

Michael D. Plummer
2006 Discussiones Mathematicae Graph Theory  
, the domination number falls to 2; (3) upper bounds on the domination number of cubic graphs; and (4) upper bounds on the domination number of graphs embedded in surfaces.  ...  In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense  ...  Additional matching results for these graphs are also discussed. In Section 4, we examine domination in cubic graphs; i.e., graphs in which each vertex has degree 3.  ... 
doi:10.7151/dmgt.1338 fatcat:4kpbebu2fff5zkdg7thtinlhhm

On 2-factors with a bounded number of odd components

Jennifer Diemunsch, Michael Ferrara, Samantha Graffeo, Timothy Morris
2014 Discrete Mathematics  
We also consider conditions that ensure the existence of a pair of disjoint 1-factors in a claw-free graph, as the union of such a pair is a 2-factor with no odd cycles.  ...  We extend results of Ryjáček, Saito, and Schelp (Closure, 2-factors, and cycle coverings in claw-free graphs, J.  ...  Theorem 5 . 5 If G is a claw-free graph, then odd(G) ≤ k if and only if odd(cl(G)) ≤ k. Corollary 1 . 1 Let G be a claw-free graph.  ... 
doi:10.1016/j.disc.2014.01.005 fatcat:xigqlmipkbbhtjz5557snq32rm

On the vertices belonging to all, some, none minimum dominating set [article]

Valentin Bouquet, François Delbot, Christophe Picouleau
2019 arXiv   pre-print
We refine this characterization for some well studied sub-classes of graphs: chordal, claw-free, triangle-free.  ...  Also we exhibit some graphs answering to some open questions of the literature on minimum dominating sets.  ...  Note that since G is (claw, P 6 )-free H is (claw,P 6 )-free too. Moreover no pair u, w of vertices of H is such that N [u] = N [w]. Figure 7 : 7 A bipartite cubic graph with γ(G) = 5.  ... 
arXiv:1909.02843v1 fatcat:3swsgwufnfg6tkvm3fu3jpr52a

Page 3572 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
Claw-free, cubic graphs of low genus have a cycle whose length is a power of two.  ...  We prove the result for cubic claw-free graphs of genus at most six and for cubic, claw-free Hamiltonian graphs of genus at most twenty-seven.” 2004e:05102 05C40 Abreu, M. (1-FLAT; Boca Raton, FL); Locke  ... 
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