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Disjoint Paired-Dominating sets in Cubic Graphs

2019
*
Graphs and Combinatorics
*

We prove that the vertex set of every

doi:10.1007/s00373-019-02063-w
pmid:31631942
pmcid:PMC6777509
fatcat:mfnujdt5dvcadhcysxyxp4tbam
*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. ...##
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How Many Conjectures Can You Stand? A Survey

2011
*
Graphs and Combinatorics
*

We survey results and open problems

doi:10.1007/s00373-011-1090-6
fatcat:jj7oxvyx5vhrzdjtv2qyj7nysy
*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*. ...##
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Locating-total domination in claw-free cubic graphs

2012
*
Discrete Mathematics
*

We show that the locating-total

doi:10.1016/j.disc.2012.06.024
fatcat:vtan26p4wne2ljpk4anow2hcua
*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 ...##
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On some intriguing problems in hamiltonian graph theory—a survey

2002
*
Discrete Mathematics
*

We survey results and open problems

doi:10.1016/s0012-365x(01)00325-9
fatcat:wcggfjuntbgrjmegqldzdz55hy
*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*. ...##
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Boxicity and cubicity of asteroidal triple free graphs

2010
*
Discrete Mathematics
*

Upper bound on boxicity of AT-

doi:10.1016/j.disc.2010.01.020
fatcat:blgsakh46jdsrodwtnp4ugcaq4
*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. ...##
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Packing 3-vertex paths in claw-free graphs and related topics

2011
*
Discrete Applied Mathematics
*

Keywords:

doi:10.1016/j.dam.2010.05.001
fatcat:74tl7ztnxrdt3ccymbrplkpbay
*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. ...##
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Boxicity and Cubicity of Asteroidal Triple free graphs
[article]

2009
*
arXiv
*
pre-print

Let G be an AT-

arXiv:0812.0894v2
fatcat:mnrbwranvbayhl3eq652xbffqi
*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*. ...##
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On a conjecture on total domination in claw-free cubic graphs

2010
*
Discrete Mathematics
*

Cockayne et al. [9] showed that every

doi:10.1016/j.disc.2010.07.006
fatcat:w5wsqvc3avhk3hfrvds3d7rfum
*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. ...##
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Packing 3-Vertex Paths in Claw-Free Graphs
[article]

2007
*
arXiv
*
pre-print

Keywords:

arXiv:0711.3871v1
fatcat:gcyai76ty5ci7p3yqyr7jja2hm
*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. ...##
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Bounds on total domination in claw-free cubic graphs

2008
*
Discrete Mathematics
*

However, every

doi:10.1016/j.disc.2007.07.007
fatcat:3ayowy544veadgrxyxlfekmvsu
*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 ...##
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Classes of cubic graphs containing cycles of integer-power lengths

2021
*
The Australasian Journal of Combinatorics
*

Erdős and Gyárfás conjectured

dblp:journals/ajc/CouchDW21
fatcat:2zq7a6qczff6pjk5ddvd4gobm4
*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*. ...##
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Some recent results on domination in graphs

2006
*
Discussiones Mathematicae Graph Theory
*

, the

doi:10.7151/dmgt.1338
fatcat:4kpbebu2fff5zkdg7thtinlhhm
*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. ...##
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On 2-factors with a bounded number of odd components

2014
*
Discrete Mathematics
*

We also consider conditions that ensure the existence of a

doi:10.1016/j.disc.2014.01.005
fatcat:xigqlmipkbbhtjz5557snq32rm
*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*. ...##
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On the vertices belonging to all, some, none minimum dominating set
[article]

2019
*
arXiv
*
pre-print

We refine this characterization for some well studied sub-classes of

arXiv:1909.02843v1
fatcat:3swsgwufnfg6tkvm3fu3jpr52a
*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. ...##
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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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