Total Roman domination for   proper interval graphs

Poureidi, Abolfazl

doi:10.5614/ejgta.2020.8.2.16

Cited by 6 publications

(5 citation statements)

References 12 publications

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“…The weight of an RDF f is f (V (G)) = u∈V (G) f (u). Roman domination was introduced by Cockayne et al in [14], and has been intensively studied in recent years [2,3,6,11,15,19].…”

Section: Introductionmentioning

confidence: 99%

Lower and upper bounds on independent double Roman domination in trees

Kheibari

Ahangar

Khoeilar

et al. 2022

EJGTA

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For a graph G = (V, E), a double Roman dominating function (DRDF) f : V → {0, 1, 2, 3} has the property that for every vertex v ∈ V with f (v) = 0, either there exists a neighbor u ∈ N (v), with f (u) = 3, or at least two neighbors x, y ∈ N (v) having f (x) = f (y) = 2, and every vertex with value 1 under f has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum f (V ) = v∈V f (v). A DRDF f is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number i dR (G) is the minimum weight of an IDRDF on G. In this paper, we show that for every tree T with diameter at least three, i(T ) + i R (T ) − s(T ) 2 + 1 ≤ i dR (T ) ≤ i(T ) + i R (T ) + s(T ) − 2, where i(T ), i R (T ) and s(T ) are the independent domination number, the independent Roman domination number and the number of support vertex of T , respectively.

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“…The weight of an RDF f is f (V (G)) = u∈V (G) f (u). Roman domination was introduced by Cockayne et al in [14], and has been intensively studied in recent years [2,3,6,11,15,19].…”

Section: Introductionmentioning

confidence: 99%

Lower and upper bounds on independent double Roman domination in trees

Kheibari

Ahangar

Khoeilar

et al. 2022

EJGTA

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“…Cockayne et al [12] introduced the concept of Roman domination in graphs, and since then many generalizations and related variations have been considered by researchers, see [7,8,9,10,11,16,19]. A variation of Roman domination, namely, double Roman domination is introduced by Beeler et al [4].…”

Section: Introductionmentioning

confidence: 99%

On the outer-independent double Italian domination number

Aziz

Kamarulhaili

Azvin

et al. 2022

EJGTA

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An outer-independent Italian dominating function (OIIDF) on a graph G is a function f : V (G) −→ {0, 1, 2} such that every vertex v ∈ V (G) with f (v) = 0 has at least two neighbors assigned 1 under f or one neighbor w with f (w) = 2, and the set {u ∈ V (G)|f (u) = 0} is independent. An outer-independent double Italian dominating function (OIDIDF) on a graph G is a functionThe minimum weight of an OIIDF (respectively, OIDIDF) on a graph G is called the outer-independent Italian (respectively, outer-independent double Italian) domination number of G. We characterize all trees T with outer-independent double Italian domination number twice the outer-independent Italian domination number. We also present lower bounds on the outer-independent double Italian domination number of a connected graph G in terms of the order, minimum and maximum degrees.

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“…The Roman dominating function for graphs has been introduced by Cockayne et al [1] and was motivated by an article in Scientific American by Ian Stewart entitled "Defend the Roman Empire" [11]. For more details on Roman domination and its variants, the reader can be referred to [2,3,8,9,12,13].…”

Section: Introductionmentioning

confidence: 99%