Bounds on the Total Restrained Domination Number of a Graph

Hattingh, Johannes H.; Jonck, Elizabeth; Joubert, Ernst J.

doi:10.1007/s00373-010-0894-0

Cited by 21 publications

(3 citation statements)

References 11 publications

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“…In [5], given two adjacent vertices u and v we say that u weakly dominates v if deg(u)≤deg(v). A set D⊆V(G) is a weak dominating set of G if every vertex in V-D is weakly dominated by atleast one vertex in D. The weak domination number γw(G) is the minimum cardinality of a weak dominating set.…”

Section: Since V[m(g)]=v(g)ue(g)mentioning

confidence: 99%

Regular Restrained Domination in Middle Graph

Muddebihal,

Mahadevappa

2023

ijmst

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In this paper, we introduce the new concept called regular restrained domination in middle graph.A set S ? V[M(G)] is a restrained dominating set if every vertex in V-S is adjacent to a vertex in S and another vertex in V-S. Note that every graph has a restrained dominating set, since S=V is such a set. Let ?rr[M(G)] denote the size of a smallest restrained dominating set. Also we study the graph theoretic properties of ?rr[M(G)] and many bounds were obtained in terms of elements of G and its relationships with other domination parameters were found.

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Section: Since V[m(g)]=v(g)ue(g)mentioning

confidence: 99%

Regular Restrained Domination in Middle Graph

Muddebihal,

Mahadevappa

2023

ijmst

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“…Similarly, we can answer many such situations using the concepts of domination. For more survey work on total restrained domination, we refer Cockayne et al [1], Cyman and Raczek [2], Hattingh et al [9][10][11], Haynes et al [12], Henning and Martiz [14], Raczek and Cyman [15], and Telle and Proskurowski [16].…”

Section: Introductionmentioning

confidence: 99%

Restrained and Total Restrained Domination of Ladder Graphs

Hemalatha,

Chandrakala,

Sooryanarayana

et al. 2023

Comm. Math. Appl.

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Telle and Proskurowksi introduced restrained domination as a vertex partition problem in partial k-tress (Algorithms for vertex partitioning problems on partial k-trees, SIAM Journal on Discrete Mathematics 10(4) (1997), 529 -550). For a graph G(V , E), a restrained domination number is the minimum cardinality of a subset D of V such that for every vertex v ∈ D there is a vertex in D as well as in D adjacent to v. If D satisfies an additional condition that every vertex of V has a neighbor in D, then D is said to be a total restrained dominating set. Minimum cardinality of D is said to be total restrained domination number of graph G. In this paper we have obtained domination, restrained, total and total restrained domination number of some ladder graphs.

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“…A minimum TOCDS of a graph G is called a γ tc (G)-set. Cyman in [1], Hattingh and Joubert in [3] obtained a lower bound for the total outer-connected domination number of a tree in terms of the order of the tree, and characterized trees achieving equality. Cyman and Raczek in [2] characterized trees with equal total domination and total outer-connected domination numbers.…”

Section: Introductionmentioning

confidence: 99%

Generalization of the total outer-connected domination in graphs

Rad¹,

Volkmann²

2016

RAIRO-Oper. Res.

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Let G = (V, E) be a graph without an isolated vertex. A set S ⊆ V is a total dominating set if S is a dominating set, and the induced subgraph G[S] does not contain an isolated vertex. The total domination number of G is the minimum cardinality of a total dominating set of G. A set D ⊆ V is a total outer-connected dominating set if D is a total dominating set, and the induced subgraph G[V − D] is connected. The total outer-connected domination number of G is the minimum cardinality of a total outer-connected dominating set of G. In this paper we generalize the total outer-connected domination number in graphs. Let k ≥ 1 be an integer. A set D ⊆ V is a total outer-k-connected component dominating set if D is a total dominating and the induced subgraph G[V − D] has exactly k connected component(s). The total outer-k-connected component domination number of G, denoted by γ k tc (G), is the minimum cardinality of a total outer-k-connected component dominating set of G. We obtain several general results and bounds for γ k tc (G), and we determine exact values of γ k tc (G) for some special classes of graphs G.

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Bounds on the Total Restrained Domination Number of a Graph

Cited by 21 publications

References 11 publications

Regular Restrained Domination in Middle Graph

Regular Restrained Domination in Middle Graph

Restrained and Total Restrained Domination of Ladder Graphs

Generalization of the total outer-connected domination in graphs

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