Strong equality of Roman and weak Roman domination in trees

Alvarado, José D.; Dantas, Simone; Rautenbach, Dieter

doi:10.1016/j.dam.2016.03.004

Cited by 11 publications

(6 citation statements)

References 11 publications

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“…Similar constructive characterizations of trees for different domination related parameters or properties have been presented in e.g. Alvarado et al (2016), Henning and Klostermeyer (2017), Henning and Marcon (2016), and Rad (2017). Our characterization is based on four simple operations.…”

Section: Corollary 6 Let G = ((A B) E G ) Be a Connected Bipartite supporting

confidence: 52%

Graphs with equal domination and covering numbers

et al. 2019

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A dominating set of a graph G is a set D ⊆ V G such that every vertex in V G − D is adjacent to at least one vertex in D, and the domination number γ (G) of G is the minimum cardinality of a dominating set of G. A set C ⊆ V G is a covering set of G if every edge of G has at least one vertex in C. The covering number β(G) of G is the minimum cardinality of a covering set of G. The set of connected graphs G for which γ (G) = β(G) is denoted by C γ =β , whereas B denotes the set of all connected bipartite graphs in which the domination number is equal to the cardinality of the smaller partite set. In this paper, we provide alternative characterizations of graphs belonging to C γ =β and B. Next, we present a quadratic time algorithm for recognizing bipartite graphs belonging to B, and, as a side result, we conclude that the algorithm of Arumugam et al. (Discrete Appl Math 161:1859-1867, 2013) allows to recognize all the graphs belonging to the set C γ =β in quadratic time either. Finally, we consider the related problem of patrolling grids with mobile guards, and show that it can be solved in O(n log n + m) time, where n is the number of line segments of the input grid and m is the number of its intersection points.

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Section: Corollary 6 Let G = ((A B) E G ) Be a Connected Bipartite supporting

confidence: 52%

Graphs with equal domination and covering numbers

et al. 2019

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“…In this paper we show that for a given graph G, it is NP-hard to decide whether γ p R (G) = γ R (G) and also in the next we provide a constructive characterization of trees T with γ p R (T ) ≡ γ R (T ). Further examples of characterizations of tress can be found in [1,2,3,4,13,16,17,19].…”

Section: Introductionmentioning

confidence: 99%

Strong equality of Roman and perfect Roman Domination in trees

et al. 2022

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A Roman dominating function (RD-function) on a graph $G = (V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. An Roman dominating function $f$ in a graph $G$ is perfect Roman dominating function (PRD-function) if every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The (perfect) Roman domination number $\gamma_R(G)$ ($\gamma_{R}^{p}(G)$) is the minimum weight of an (perfect) Roman dominating function on $G$. We say that $\gamma_{R}^{p}(G)$ strongly equals $\gamma_R(G)$, denoted by $\gamma_{R}^{p}(G)\equiv \gamma_R(G)$, if every RD-function on $G$ of minimum weight is a PRD-function. In this paper we show that for a given graph $G$, it is NP-hard to decide whether $\gamma_{R}^{p}(G)= \gamma_R(G)$ and also we provide a constructive characterization of trees $T$ with $\gamma_{R}^{p}(T)\equiv \gamma_R(T)$.

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“…The weight of an RDF f is w( f ) = f (V(G)) = ∑ u∈V(G) f (u). We denote by γ R (G) the minimum weight of an RDF of G. It is called the the Roman domination number of G. For some further results on Roman dominating function in graphs, we recommend the reader to consult the papers [4][5][6][7]. RDFs are useful in the study of generalized k-core percolation [8], where V 2 = α-removable, V 0 = β-removable, and V 1 = non-removable vertices.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, finding the weak Roman domination number for some special classes of graphs as well as finding some good bounds for this invariant is of great importance. For some recent results on the weak Roman domination number of graphs we refer the interested reader to the papers [4,13].…”

Section: Introductionmentioning

confidence: 99%

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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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Strong equality of Roman and weak Roman domination in trees

Cited by 11 publications

References 11 publications

Graphs with equal domination and covering numbers

Graphs with equal domination and covering numbers

Strong equality of Roman and perfect Roman Domination in trees

Strong Equality of Perfect Roman and Weak Roman Domination in Trees

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