<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:math>-domination in graphs

Yang, Xiaojing; Wu, Baoyindureng

doi:10.1016/j.dam.2014.05.035

Cited by 39 publications

(24 citation statements)

References 3 publications

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“…Recently, a variation of the domination problem, called [a,b]-set, was proposed and studied (Chellali, Haynes, Hedetniemi & McRae, 2013;Yang & Wu, 2014;Goharshady, Hooshmandasl & Meybodi,2016). A vertex subset S of a graph G = (V, E) is an [a, b]-set if, a ≤ |N(v) ∩ S | ≤ b for every vertex v ∈ V \ S , that is, each vertex v ∈ V \ S is adjacent to either one or two vertices in S .…”

Section: Let G Be a Graph S ⊆ V(g) V ∈ V(g) The Open Neighborhood mentioning

confidence: 99%

The [a,b]-domination and [a,b]-total Domination of Graphs

Zhang¹,

Shao²,

Hong³

2017

JMR

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Section: Let G Be a Graph S ⊆ V(g) V ∈ V(g) The Open Neighborhood mentioning

confidence: 99%

The [a,b]-domination and [a,b]-total Domination of Graphs

Zhang¹,

Shao²,

Hong³

2017

JMR

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“…To the best of our knowledge, only three papers have appeared on (i, j)-set or its variants [6,19,8]. The main focus of [6] is on a particular (i, j)-set, namely (1, 2)-set.…”

Section: (I J)-set Problem ((I J)-set)mentioning

confidence: 99%

“…A list of open problems were posed in [6], some of which were solved in [19]. In [19], the authors showed that there exist planar and bipartite graphs with γ (1,2) (G) = n. They also showed that for a tree T with k leaves, if deg G (v) ≥ 4 for any non-leaf vertex v, then γ (1,2) (T ) = n − k. Nordhaus-Gaddum-type inequalities are also established for (1, 2)-set in [19]. In [8], quasiperfect domination has been studied in triangular lattices.…”

Section: (I J)-set Problem ((I J)-set)mentioning

confidence: 99%

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(1,j)-set problem in graphs

Bishnu

Dutta

Ghosh

et al. 2016

Discrete Mathematics

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a b s t r a c tA subset D ⊆ V of a graph G = (V , E) is a (1, j)-set (Chellali et al., 2013) if every vertex v ∈ V \D is adjacent to at least 1 but not more than j vertices in D. The cardinality of a minimum (1, j)-set of G, denoted as γ (1,j) (G), is called the (1, j)-domination number of G. In this paper, using probabilistic methods, we obtain an upper bound on γ (1,j) (G) for j ≥ O(log ∆), where ∆ is the maximum degree of the graph. The proof of this upper bound yields a randomized linear time algorithm. We show that the associated decision problem is NP-complete for choral graphs but, answering a question of Chellali et al., provide a linear-time algorithm for trees for a fixed j. Apart from this, we design a polynomial time algorithm for finding γ (1,j) (G) for a fixed j in a split graph, and show that (1, j)-set problem is fixed parameter tractable in bounded genus graphs and bounded treewidth graphs.

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“…Note that γ [1,2] (G) ≤ n is a trivial upper bound for a graph G of order n, and the equality holds for an infinite many values of n, see [4,14]. In the following theorem, we establish a sharp upper bound for the total [1,2]-domination number of a connected graph in terms of its order and characterize all graphs achieving the bound.…”

mentioning

confidence: 97%