On perfect Roman domination number in trees: complexity and bounds

Darkooti, Mahsa; Alhevaz, Abdollah; Rahimi, Sadegh; Rahbani, Hadi

doi:10.1007/s10878-019-00408-y

Cited by 7 publications

(4 citation statements)

References 7 publications

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“…Similarly, perfect Roman domination number γ pR (G) is defined. In [5], Henning et al introduced the notion of perfect Roman domination and showed that if T is a tree on n ≥ 3 vertices, then γ pR (T ) ≤ 4 5 n. In [3], Darkooti et al proved that it is NP-complete to decide whether a graph has a perfect Roman dominating function, even if the graph is bipartite. This suggests determining the exact value of perfect Roman domination numbers for special classes of graphs.…”

Section: Introductionmentioning

confidence: 99%

Perfect Roman domination in middle graphs

Kim¹

2021

Preprint

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The middle graph M (G) of a graph G is the graph obtained by subdividing each edge of G exactly once and joining all these newly introduced vertices of adjacent edges of G. A perfect Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex v with f (v) = 0 is adjacent to exactly one vertex u for which f (u) = 2. The weight of a perfect Roman dominating function f is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on G. In this paper, we give a characterization of middle graphs with equal Roman domination and perfect Roman domination numbers.

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Section: Introductionmentioning

confidence: 99%

Perfect Roman domination in middle graphs

Kim¹

2021

Preprint

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“…This is because (∅, V(G), ∅) is a PRDF for G. Note also that every graph G of order n satisfies γ P R (G) ≤ n, as one can define a perfect Roman dominating function f by letting f (u) = 1 for any vertex u of G. A PRDF of G with minimum weight is also called a γ P R (G)-function (see Figure 1). For some recent results on perfect Roman domination number of graphs we refer the readers to the papers [10,11]. For graph G = (V, E) and function f : V −→ {0, 1, 2}, a vertex u with f (u) = 0 is undefended with respect to f if all its neighbors have only zero weight.…”

Section: Introductionmentioning

confidence: 99%

“…Accordingly, we must have equalities throughout the inequality chain (10). In particular, γ r (T) = γ P R (T).…”

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confidence: 99%

Strong Equality of Perfect Roman and Weak Roman Domination in Trees

et al. 2019

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Let G = ( V , E ) be a graph and f : V ⟶ { 0 , 1 , 2 } be a function. Given a vertex u with f ( u ) = 0 , if all neighbors of u have zero weights, then u is called undefended with respect to f. Furthermore, if every vertex u with f ( u ) = 0 has a neighbor v with f ( v ) > 0 and the function f ′ : V ⟶ { 0 , 1 , 2 } with f ′ ( u ) = 1 , f ′ ( v ) = f ( v ) − 1 , f ′ ( w ) = f ( w ) if w ∈ V ∖ { u , v } has no undefended vertex, then f is called a weak Roman dominating function. Also, the function f is a perfect Roman dominating function if every vertex u with f ( u ) = 0 is adjacent to exactly one vertex v for which f ( v ) = 2 . Let the weight of f be w ( f ) = ∑ v ∈ V f ( v ) . The weak (resp., perfect) Roman domination number, denoted by γ r ( G ) (resp., γ R p ( G ) ), is the minimum weight of the weak (resp., perfect) Roman dominating function in G. In this paper, we characterize those trees where the perfect Roman domination number strongly equals the weak Roman domination number, in the sense that each weak Roman dominating function of minimum weight is, at the same time, perfect Roman dominating.

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“…A PIDF f is called a γ p I (G)-function if ω(f ) = γ p I (G). It was shown in [10] that every tree T of order n ≥ 3 satisfies γ p R (T ) ≤ 4 5 n. However, this upper bound has recently been improved by Darkooti et al [8] for trees T with (T ) ≥ 2s(T ) − 2, by showing that for any tree T of order n ≥ 3 with (T ) leaves and s(T ) support vertices, γ p R (T ) ≤ (4n− (T ) + 2s(T ) − 2)/5. Moreover, Henning and Haynes showed in [9] that 4 5 n is also an upper bound of the prefect Italian domination number for any tree of order n ≥ 3.…”

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confidence: 99%

A new upper bound for the perfect Italian domination number of a tree

Nazari-Moghaddam¹,

Chellali²

2020

Discuss. Math. Graph Theory

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A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f (u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γ p I (G), is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with (T ) leaves and s(T ) support vertices, γ p I (T ) ≤ 4n− (T )+2s(T )−1 5 , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164-177].

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On perfect Roman domination number in trees: complexity and bounds

Cited by 7 publications

References 7 publications

Perfect Roman domination in middle graphs

Perfect Roman domination in middle graphs

Strong Equality of Perfect Roman and Weak Roman Domination in Trees

A new upper bound for the perfect Italian domination number of a tree

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