Roman Domination in Graphs

Chellali, Mustapha; Rad, Nader Jafari; Sheikholeslami, Seyed Mahmoud; Volkmann, Lutz

doi:10.1007/978-3-030-51117-3_11

Cited by 46 publications

(8 citation statements)

References 45 publications

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“…The weight of an RDF f is the value f (V (G)) = u∈V (G) f (u), and the Roman domination number γ R (G) is the minimum weight of an RDF on G. The Roman domination number of graphs was introduced in 2004 by Cockayne et al in [14] and is now well-studied in graph theory. The literature on Roman domination and its variations has been surveyed and detailed in two book chapters and three surveys [9,10,11,12,13].…”

Section: Conjecture 1 ([1]mentioning

confidence: 99%

Roman domination in graphs with minimum degree at least two and some forbidden cycles

Sheikholeslami¹,

Chellali²,

Khoeilar³

et al. 2021

Preprint

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Let G = (V, E) be a graph of order n and let γ R (G) and ∂(G) denote the Roman domination number and the differential of G, respectively. In this paper we prove that for any integer k ≥ 0, if G is a graph of order n ≥ 6k + 9, minimum degree δ ≥ 2, which does not contain any induced {C 5 , C 8 , . . . , C 3k+2 }-cycles, then γ R (G) ≤ (4k+8)n 6k+11 . This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N.

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Section: Conjecture 1 ([1]mentioning

confidence: 99%

Roman domination in graphs with minimum degree at least two and some forbidden cycles

Sheikholeslami¹,

Chellali²,

Khoeilar³

et al. 2021

Preprint

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“…Hedetniemi and S.T. Hedetniemi [10] introduced the concept of Roman domination in graphs, and since then a lot of related variations and generalizations have been studied (see [6][7][8][9]). In 2016, Beeler, Haynes and S.T.…”

Section: Introductionmentioning

confidence: 99%

Double Roman and double Italian domination

Volkmann¹

2023

Discuss. Math. Graph Theory

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Let G be a graph with vertex set V (G). A double Roman dominating function (DRDF) on a graph G is a function f : V (G) −→ {0, 1, 2, 3} that satisfies the following conditions: (i) If f (v) = 0, then v must have a neighbor w with f (w) = 3 or two neighbors x and y with f (x) = f (y) = 2; (ii) If f (v) = 1, then v must have a neighbor w with f (w) ≥ 2. The weight of a DRDF f is the sum v∈V (G) f (v). The double Roman domination number equals the minimum weight of a double Roman dominating function on G. A double Italian dominating function (DIDF) is a function f : V (G) −→ {0, 1, 2, 3} having the property that f (N [u]) ≥ 3 for every vertex u ∈ V (G) with f (u) ∈ {0, 1}, where N [u] is the closed neighborhood of v. The weight of a DIDF f is the sum v∈V (G) f (v), and the minimum weight of a DIDF in a graph G is the double Italian domination number. In this paper we first present Nordhaus-Gaddum type bounds on the double Roman domination number which improved corresponding results given in [N. Jafari Rad and H. Rahbani, Some progress on the double Roman domination in graphs, Discuss. Math. Graph Theory 39 (2019) 41-53]. Furthermore, we establish lower bounds on the double Roman and double Italian domination numbers of trees.

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“…Extended forms of domination in graphs have been vast in the domain literature. Some very recent forms include broadcast domination [18], pitchfork domination [19], Roman domination [20], double Roman domination [21], triple Roman domination [22], captive domination [23], outer-convex domination [24], and paired domination [25], among others. The trajectory of these topics has been exponential in the last decade.…”

Section: Introductionmentioning

confidence: 99%

Domination in Fuzzy Directed Graphs

et al. 2021

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A new domination parameter in a fuzzy digraph is proposed to espouse a contribution in the domain of domination in a fuzzy graph and a directed graph. Let GD*=V,A be a directed simple graph, where V is a finite nonempty set and A=x,y:x,y∈V,x≠y. A fuzzy digraph GD=σD,μD is a pair of two functions σD:V→0,1 and μD:A→0,1, such that μDx,y≤σDx∧σDy, where x,y∈V. An edge μDx,y of a fuzzy digraph is called an effective edge if μDx,y=σDx∧σDy. Let x,y∈V. The vertex σDx dominates σDy in GD if μDx,y is an effective edge. Let S⊆V, u∈V\S, and v∈S. A subset σDS⊆σD is a dominating set of GD if, for every σDu∈σD\σDS, there exists σDv∈σDS, such that σDv dominates σDu. The minimum dominating set of a fuzzy digraph GD is called the domination number of a fuzzy digraph and is denoted by γGD. In this paper, the concept of domination in a fuzzy digraph is introduced, the domination number of a fuzzy digraph is characterized, and the domination number of a fuzzy dipath and a fuzzy dicycle is modeled.

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Roman Domination in Graphs

Cited by 46 publications

References 45 publications

Roman domination in graphs with minimum degree at least two and some forbidden cycles

Roman domination in graphs with minimum degree at least two and some forbidden cycles

Double Roman and double Italian domination

Domination in Fuzzy Directed Graphs

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