The Signed Roman Domatic Number of a Digraph

Let $D$ be a finite and simple digraph with vertex set $V(D)$.A {\em signed Roman dominating function} on the digraph $D$ isa function $f:V (D)\longrightarrow \{-1, 1, 2\}$ such that$\sum_{u\in N^-[v]}f(u)\ge 1$ for every $v\in V(D)$, where $N^-[v]$ consists of $v$ andall inner neighbors of $v$, and every vertex $u\in V(D)$ for which $f(u)=-1$ has an innerneighbor $v$ for which $f(v)=2$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signedRoman dominating functions on $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$ for each$v\in V(D)$, is called a {\em signed Roman dominating family} (of functions) on $D$. The maximumnumber of functions in a signed Roman dominating family on $D$ is the {\em signed Roman domaticnumber} of $D$, denoted by $d_{sR}(D)$. In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for $d_{sR}(D)$. 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

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

Total kk-Rainbow domination numbers in graphs

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