ON PAIRED-DOUBLE DOMINATION NUMBER OF GRAPHS

A vertex subset S of a graph G = (V, E) is a double dominating set for G if |N[v]∩S| ≥ 2 for each vertex v ∈ V, where N[v] = {u |uv ∈ E}∪{v}. The double domination number of G, denoted by γ×2(G), is the cardinality of a smallest double dominating set of G. A graph G is said to be double domination edge critical if γ×2(G + e) < γ×2(G) for any edge e ∉ E. A double domination edge critical graph G with γ×2(G) = k is called k - γ×2(G)-critical. In this paper, we first show that G has a perfect matching if G is a connected 3 - γ×2(G)-critical graph of even order. Secondly, we show that G is factor-critical if G is a connected 3 - γ×2(G)-critical graph with odd order and minimum degree at least 2. Finally, we show that G is factor-critical if G is a connected K1,4-free 4 - γ×2(G)-critical graph of odd order with minimum degree at least 2.

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On Double Domination Numbers of Generalized Petersen Graphs

Journal of Computational and Theoretical Nanoscience ◽

10.1166/jctn.2016.5595 ◽

2016 ◽

Vol 13 (10) ◽

pp. 6514-6518

Author(s):

Minhong Sun ◽

Zehui Shao

Keyword(s):

Programming Model ◽

Dominating Set ◽

Domination Number ◽

Upper Bounds ◽

Minimum Size ◽

Generalized Petersen Graphs ◽

Small Parameters ◽

Petersen Graphs ◽

Double Domination ◽

Domination Numbers

A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The result shows that the given upper bounds match these exact values.

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A new lower bound on the double domination number of a graph

Discrete Applied Mathematics ◽

10.1016/j.dam.2018.06.009 ◽

2019 ◽

Vol 254 ◽

pp. 280-282 ◽

Cited By ~ 2

Author(s):

Majid Hajian ◽

Nader Jafari Rad

Keyword(s):

Lower Bound ◽

Domination Number ◽

Double Domination

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Double domination and super domination in trees

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830916500671 ◽

2016 ◽

Vol 08 (04) ◽

pp. 1650067

Author(s):

B. Krishnakumari ◽

Y. B. Venkatakrishnan

Keyword(s):

Dominating Set ◽

Domination Number ◽

Minimum Cardinality ◽

Double Domination

A vertex of a graph [Formula: see text] is said to dominate itself and all its neighbors. A double dominating set (DDS) of a graph [Formula: see text] is a set [Formula: see text] of vertices such that every vertex of [Formula: see text] is dominated by at least two vertices of [Formula: see text]. The double domination number of a graph [Formula: see text] is the minimum cardinality of a DDS of [Formula: see text]. For a graph [Formula: see text], a subset [Formula: see text] of [Formula: see text] is a super dominating set SDS if for every vertex of [Formula: see text] there exists an external private neighbor of [Formula: see text] with respect to [Formula: see text]. The super domination number of [Formula: see text] is the minimum cardinality of a SDS of [Formula: see text]. We prove that for every tree [Formula: see text], [Formula: see text], and we characterize the trees attaining this bound.

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An upper bound on the double domination number of trees

Kragujevac Journal of Mathematics ◽

10.5937/kgjmath1502133a ◽

2015 ◽

Vol 39 (2) ◽

pp. 133-139 ◽

Cited By ~ 1

Author(s):

J. Amjadi

Keyword(s):

Upper Bound ◽

Domination Number ◽

Double Domination

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Double Domination Number of Some Families of Graph

International Journal of Recent Technology and Engineering - 2 ◽

10.35940/ijrte.b3307.079220 ◽

2020 ◽

Vol 9 (2) ◽

pp. 161-164

Keyword(s):

Maximum Degree ◽

Dominating Set ◽

Domination Number ◽

Ladder Graph ◽

Closed Neighborhood ◽

Circular Ladder ◽

Double Domination

In a graph G = (V, E) each vertex is said to dominate every vertex in its closed neighborhood. In a graph G, if S is a subset of V then S is a double dominating set of G if every vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double domination number γx2 (G). [4]. In this paper, we computed some relations between double domination number, domination number, number of vertices (n) and maximum degree (∆) of Helm graph, Friendship graph, Ladder graph, Circular Ladder graph, Barbell graph, Gear graph and Firecracker graph.

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