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A note on dominating cycles in 2-connected graphs

D. Bauer, E. Schmeichel, H.J. Veldman
1996 Discrete Mathematics  
We prove that every longest cycle in G is a dominating cycle unless G is a spanning subgraph of a graph belonging to one of four easily specified classes of graphs.  ...  Let G be a 2-connected graph on n vertices such that d(x)+ d(y)+ d(z)>/n for all triples of independent vertices x,y,z.  ...  Let G be an Ogl-tOugh graph on n vertices with 63 ~ n. Then every longest cycle in G is a dominating cycle.  ... 
doi:10.1016/0012-365x(94)00364-o fatcat:nrlvfrmodbcqhliq3xg2kr3joa

Edge-dominating cycles, k-walks and Hamilton prisms in 2K2-free graphs

Gao Mou, Dmitrii V. Pasechnik
2016 Journal of knot theory and its ramifications  
We show that an edge-dominating cycle in a 2K_2-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough 2K_2-free graph admits a k-walk, and it can be found in polynomial time  ...  Furthermore, we prove that for any ϵ>0 every (1+ϵ)-tough 2K_2-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other  ...  The authors thank Nick Gravin and Edith Elkind for helpful comments on drafts of this text. Research supported by Singapore MOE Tier 2 Grant MOE2011-T2-1-090 (ARC 19/11).  ... 
doi:10.1142/s0218216516420116 fatcat:sg2cd7pz75ghbkb6sdunvomvve

On the length of longest dominating cycles in graphs

Hoa Vu Dinh
1993 Discrete Mathematics  
A graph G is called cycle-dominable if G contains a dominating cycle. There exists l-tough graph in which no longest cycle is dominating.  ...  ., On the length of longest dominating cycles in graphs, Discrete Mathematics 121 (1993) 21 l-222. A cycle C in an undirected and simple graph G is dominating if G -C is edgeless.  ...  Note that there are graphs containing a dominating cycle in which no longest cycle is dominating.  ... 
doi:10.1016/0012-365x(93)90554-7 fatcat:wqjo6awccbhbfj7cgmrq7a7ox4

Page 723 of Mathematical Reviews Vol. , Issue 98B [page]

1998 Mathematical Reviews  
Sampathkumar [“Recent developments in the theory of domination in graphs”, MRI Lecture Notes No. 1, Allahabad, 1979; per bibl.].  ...  In this note graphs having a set consisting of any pair of vertices of the graph as a minimal dominating set are characterised. This also settles a conjecture of H. B. Walikar, B. D. Acharya and E.  ... 

Toughness in Graphs – A Survey

Douglas Bauer, Hajo Broersma, Edward Schmeichel
2006 Graphs and Combinatorics  
A dominating cycle of G is a cycle C of G such that G − V (C) is an independent set, i.e., such that every edge of G has at least one of its endvertices on C.  ...  Toughness and Circumference In this section we survey results concerning the relationship between the toughness of a graph and its circumference.  ...  Let G be a 1-tough graph on n vertices with δ ≥ n/3. Then every longest cycle in G is a dominating cycle. Theorem 15. Let G be a 1-tough graph on n ≥ 3 vertices with δ ≥ max{n/3, α − 1}.  ... 
doi:10.1007/s00373-006-0649-0 fatcat:pybxyua33rehpmiybdbj2cnzru

Long cycles in graphs with large degree sums

Douglas Bauer, H.J. Veldman, A. Morgana, E.F. Schmeichel
1990 Discrete Mathematics  
We close by noting that an application of Theorem 7 and Lemma 8 leads to a simple proof of Jung's Theorem for graphs on at least 16 vertices.  ...  The graph obtained from HI3 by deleting a vertex of degree 4 shows that the requirement that n 2 13 cannot be released.  ...  Theorem 6 . 6 Let G be a l-tough graph on n vertices with 6 2 in. Then every longest cycle in G is a dominating cycle.  ... 
doi:10.1016/0012-365x(90)90055-m fatcat:kqstuqqrijgbvd5ymarp653kxq

Polynomial algorithms that prove an NP-Hard hypothesis implies an NP-hard conclusion

D. Bauer, H.J. Broersma, A. Morgana, E. Schmeichel
2002 Discrete Applied Mathematics  
In this note we point out that other theorems in hamiltonian graph theory have a similar character. In particular, we present a constructive proof of a well-known theorem of Jung (Ann.  ...  A number of results in hamiltonian graph theory are of the form "P1 implies P2", where P1 is a property of graphs that is NP-hard and P2 is a cycle structure property of graphs that is also NP-hard.  ...  Let G be a graph on n vertices with 3 ¿ n. If G is 1-tough; then every longest cycle in G is a dominating cycle.  ... 
doi:10.1016/s0166-218x(01)00276-1 fatcat:lbdcybfk2fcgdehtu5wdfqoopq

Author index to volume

2004 Discrete Mathematics  
Tuza, On short cycles through prescribed vertices of a graph (1-2) 67-74 G . orlich, A., M. Pil! sniak, M. Wo! zniak and I.A. Zio"o, A note on embedding graphs without short cycles (1-2) 75-77 G !  ...  zniak, A note on pancyclism of highly connected graphs (1-2) 57-60 Flandrin, E., see M.E.K. Abderrezzak (1-2) 5-13 Fletcher, R.J., C. Koukouvinos and J.  ...  Tuza, On short cycles through prescribed vertices of a graph (1-2) 67-74 G . orlich, A., M. Pil! sniak, M. Wo! zniak and I.A. Zio"o, A note on embedding graphs without short cycles (1-2) 75-77 G !  ... 
doi:10.1016/s0012-365x(04)00345-0 fatcat:6zribkcwoneppcmbtj3xqfm3qy

A simple proof of a theorem of Jung

Douglas Bauer, Aurora Morgana, E.F. Schmeichel
1990 Discrete Mathematics  
Jung's theorem states that if G is a l-tough graph on n 3 11 vertices such that d(x) + d(y) 2 n -4 for all distinct nonadjacent vertices x, y, then G is hamiltonian.  ...  We give a simple proof of this theorem for graphs with 16 or more vertices.  ...  A cycle C of G is a dominating cycle if every edge of G has at least one of its vertices on C.  ... 
doi:10.1016/0012-365x(90)90029-h fatcat:c2yv4gs2wrdiboi2oputs6s7xy

A note on interconnecting matchings in graphs

Tomáš Kaiser
2006 Discrete Mathematics  
We apply the result to the problem of the existence of a (spanning) 2-walk in sufficiently tough graphs.  ...  We prove a sufficient condition for a graph G to have a matching that interconnects all the components of a disconnected spanning subgraph of G.  ...  [8] , every 2-tough graph contains a 2-factor; let F be a 2-factor in G.  ... 
doi:10.1016/j.disc.2006.05.011 fatcat:dvbire6rsbdodegzghjfhs5qdi

Long cycles in graphs with prescribed toughness and minimum degree

Douglas Bauer, H.J. Broersma, J. van den Heuvel, H.J. Veldman
1995 Discrete Mathematics  
Using the notion of D~-cycles, a number of results are established concerning long cycles in graphs with prescribed toughness and minimum degree. Let G be a t-tough graph on n/> 3 vertices.  ...  A cycle C of a graph G is a D~-cycle if every component of G-V(C) has order less than 2.  ...  Let G be a 1-tough graph on n>~3 vertices with >~n. Then G contains a dominating cycle.  ... 
doi:10.1016/0012-365x(93)e0204-h fatcat:eted23xqqfde5owcjmc7p5nwce

2-walks in 2-tough 2k2-free graphs [article]

Gao Mou
2014 arXiv   pre-print
In this paper, we prove that in every 2-tough 2K_2-free graph, there is a 2-walk.  ...  Every 1/(k − 2)-tough graph has a k-walk. In particular every 1-tough graph has a 3-walk.  ...  For some classes of graphs, there are strong results connected toughness and Hamiltonicity (recall that a 1-walk is a Hamiltonian cycle). E.g. in [2] , G. Chen, M.S. Jackson, A.E. Kezdy, and J.  ... 
arXiv:1408.3380v3 fatcat:pcrwqjzmfrcbhgron6iti3mfsy

Vertex-dominating cycles in 2-connected bipartite graphs

Tomoki Yamashita
2007 Discussiones Mathematicae Graph Theory  
In this paper, we prove that if G is a 2-connected bipartite graph with partite sets V 1 and V 2 such that δ(G) ≥ 1 3 (max{|V 1 |, |V 2 |} + 1), then G has a vertex-dominating cycle.  ...  A cycle C is a vertex-dominating cycle if every vertex is adjacent to some vertex of C. Bondy and Fan [4] − 4) , then G has a vertex-dominating cycle.  ...  For example, Jung (1972) and Moon and Moser (1963) showed that weaker degree sum conditions guarantee hamiltonian cycles in 1-tough graphs and in bipartite graphs, respectively.  ... 
doi:10.7151/dmgt.1364 fatcat:qtktpzttrne6bhyuuljwdod6ba

Research problems from the 18th Workshop '3in1' 2009

Zdeněk Ryjáček, Carol T. Zamfirescu
2010 Opuscula Mathematica  
A collection of open problems that were posed at the 18th Workshop '3in1', held on November 26-28, 2009 in Krakow, Poland.  ...  The problems are presented by Zdenek Ryjacek in "Does the Thomassen's conjecture imply N=NP?" and "Dominating cycles and hamiltonian prisms", and by Carol T.  ...  A dominating cycle in a graph G is a cycle C such that every edge of G has at least one vertex on C, i.e. such that the graph G − C is edgeless.  ... 
doi:10.7494/opmath.2010.30.4.527 fatcat:raomxgubl5a5ppcrbmhgqyza6y

Graph Invariants and Large Cycles: A Survey

Zh. G. Nikoghosyan
2011 International Journal of Mathematics and Mathematical Sciences  
In this paper, we have collected 36 pure algebraic relations between basic (initial) graph invariants ensuring the existence of a certain type of large cycles.  ...  This paper is devoted to large cycle substructures, namely, Hamilton, longest and dominating cycles and some generalized cycles including Hamilton and dominating cycles as special cases.  ...  Then each longest cycle in G is a dominating cycle. Theorem 4. 4 4 Yamashita 22 . Let G be graph with κ ≥ 3 and δ ≥ n κ 3 /4. Then each longest cycle in G is a dominating cycle.  ... 
doi:10.1155/2011/206404 fatcat:gnubm2vu6vanrlvgyrlt7sa42q
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