24 research outputs found

    The Signed Roman Domatic Number of a Digraph

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    Let DD be a finite and simple digraph with vertex set V(D)V(D).A {\em signed Roman dominating function} on the digraph DD isa function f:V(D)⟶{−1,1,2}f:V (D)\longrightarrow \{-1, 1, 2\} such that∑u∈N−[v]f(u)≥1\sum_{u\in N^-[v]}f(u)\ge 1 for every v∈V(D)v\in V(D), where N−[v]N^-[v] consists of vv andall inner neighbors of vv, and every vertex u∈V(D)u\in V(D) for which f(u)=−1f(u)=-1 has an innerneighbor vv for which f(v)=2f(v)=2. A set {f1,f2,…,fd}\{f_1,f_2,\ldots,f_d\} of distinct signedRoman dominating functions on DD with the property that ∑i=1dfi(v)≤1\sum_{i=1}^df_i(v)\le 1 for eachv∈V(D)v\in V(D), is called a {\em signed Roman dominating family} (of functions) on DD. The maximumnumber of functions in a signed Roman dominating family on DD is the {\em signed Roman domaticnumber} of DD, denoted by dsR(D)d_{sR}(D). In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for dsR(D)d_{sR}(D). In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs

    Signed total double Roman dominatıon numbers in digraphs

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    Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio

    Annales Mathematicae et Informaticae (38.)

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    Annales Mathematicae et Informaticae (40.)

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    Total kk-Rainbow domination numbers in graphs

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    Let k≥1k\geq 1 be an integer‎, ‎and let GG be a graph‎. ‎A {\it‎ ‎kk-rainbow dominating function} (or a {\it kk-RDF}) of GG is a‎ ‎function ff from the vertex set V(G)V(G) to the family of all subsets‎ ‎of {1,2,…‎,‎k}\{1,2,\ldots‎ ,‎k\} such that for every v∈V(G)v\in V(G) with‎ ‎f(v)=∅f(v)=\emptyset ‎, ‎the condition ⋃u∈‎‎NG(v)f(u)={1,2,…,k}\bigcup_{u\in‎ ‎N_{G}(v)}f(u)=\{1,2,\ldots,k\} is fulfilled‎, ‎where NG(v)N_{G}(v) is‎ ‎the open neighborhood of vv‎. ‎The {\it weight} of a kk-RDF ff of‎ ‎GG is the value ω(f)=∑v∈V(G)∣f(v)∣\omega (f)=\sum _{v\in V(G)}|f(v)|‎. ‎A kk-rainbow‎ ‎dominating function ff in a graph with no isolated vertex is called‎ ‎a {\em total kk-rainbow dominating function} if the subgraph of GG‎ ‎induced by the set {v∈V(G)∣f(v)≠∅}\{v \in V(G) \mid f (v) \not =\emptyset\} has no isolated‎ ‎vertices‎. ‎The {\em total kk-rainbow domination number} of GG‎, ‎denoted by‎ ‎γtrk(G)\gamma_{trk}(G)‎, ‎is the minimum weight of a total kk-rainbow‎ ‎dominating function on GG‎. ‎The total 11-rainbow domination is the‎ ‎same as the total domination‎. ‎In this paper we initiate the‎ ‎study of total kk-rainbow domination number and we investigate its‎ ‎basic properties‎. ‎In particular‎, ‎we present some sharp bounds on the‎ ‎total kk-rainbow domination number and we determine the total‎ ‎kk-rainbow domination number of some classes of graphs‎.
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