On total restrained domination in graphs

Ma, Dengke; Chen, Xue-gang; Sun, Liang

doi:10.1007/s10587-005-0012-2

Cited by 34 publications

(7 citation statements)

References 4 publications

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“…The total restrained domination number of , denoted by TR ( ), is the minimum cardinality of a total restrained dominating set of . For references on this domination in graphs, see [3,61,[82][83][84][85]. The total restrained bondage number TR ( ) of a graph with no isolated vertex, is the cardinality of a smallest set of edges ⊆ ( ) for which − has no isolated vertex and TR ( − ) > TR ( ).…”

Section: Corollary 134 the Bondage Number And The Restrained Bondagementioning

confidence: 99%

“…In the case that there is no such subset , we define TR ( ) = ∞. Ma et al [84] determined that TR ( ) = − 2⌊( − 2)/4⌋ for ⩾ 3, TR ( ) = − 2⌊ /4⌋ for ⩾ 3, TR ( 3 ) = 3 and TR ( ) = 2 for ̸ = 3, and TR ( 1, ) = 1 + and TR ( , ) = 2 for 2 ⩽ ⩽ . According to these results, Jafari Rad et al [81] determined the exact values of total restrained bondage numbers for the corresponding graphs.…”

Section: Corollary 134 the Bondage Number And The Restrained Bondagementioning

confidence: 99%

See 1 more Smart Citation

On Bondage Numbers of Graphs: A Survey with Some Comments

2013

International Journal of Combinatorics

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The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of . The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of . The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers.

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Section: Corollary 134 the Bondage Number And The Restrained Bondagementioning

confidence: 99%

Section: Corollary 134 the Bondage Number And The Restrained Bondagementioning

confidence: 99%

On Bondage Numbers of Graphs: A Survey with Some Comments

2013

International Journal of Combinatorics

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“…The total domination number of G, denoted by γ t (G), is the minimum cardinality of a TDS of G, while the total restrained domination number of G, denoted by γ tr (G), is the minimum cardinality of a TRDS of G. Total domination in graphs is very well studied in graph theory (see, for example, the recent papers [8,[10][11][12]19]). Recent papers on total restrained domination in graphs can be found, for example, in [2,5,6,9,13,[15][16][17]21].…”

Section: Introductionmentioning

confidence: 99%

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

Southey

Henning

2011

Graphs and Combinatorics

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In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of V \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ tr (G) of G. Jiang et al. (Graphs Combin 25:341-350, 2009) showed that if G is a connected cubic graph of order n, then γ tr (G) ≤ 13n/19. In this paper we improve this upper bound to γ tr (G) ≤ (n + 4)/2. We provide two infinite families of connected cubic graphs G with γ tr (G) = n/2, showing that our new improved bound is essentially best possible.

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“…Total domination in graphs was introduced by Cockayne, Dawes, and Hedetniemi [3] and further studied, for example, in [1,8,9,16], while restrained domination was introduced by Telle and Proskurowski [18], albeit indirectly, as a vertex partitioning problem and further studied, for example, in [5][6][7]11,15]. The concept of total restrained domination in graphs was also introduced in [18], albeit indirectly, as a vertex partitioning problem and has been studied, for example, in [12,17,19].…”

Section: Introductionmentioning

confidence: 99%

Total restrained domination in graphs with minimum degree two

Henning

Maritz

2008

Discrete Mathematics

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In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) as a vertex partitioning problem. A set S of vertices in a graph G = (V , E) is a total restrained dominating set of G if every vertex is adjacent to a vertex in S and every vertex of V \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by tr (G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree where n − 2. We prove that if n 4, then tr (G) n − 2 − 1 and this bound is sharp. If we restrict G to a bipartite graph with 3, then we improve this bound by showing that tr (G) n − 2 3 − 2 9 √ 3 − 8 − 7 9 and that this bound is sharp.

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On total restrained domination in graphs

Cited by 34 publications

References 4 publications

On Bondage Numbers of Graphs: A Survey with Some Comments

On Bondage Numbers of Graphs: A Survey with Some Comments

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

Total restrained domination in graphs with minimum degree two

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