Signed star k-domatic number of a graph

Let $G$ be a simple graph without isolated vertices with vertex set $V(G)$ and edge set $E(G)$ and let $k$ be a positive integer. A function $f:E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star $k$-dominating function on $G$ if $\sum_{e\in E(v)}f(e)\ge k$ for every vertex $v$ of $G$, where $E(v)=\{uv\in E(G)\mid u\in N(v)\}$. A set $\{f_1,f_2,\ldots,f_d\}$ of signed star $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(e)\le 1$ for each $e\in E(G)$, is called a signed star $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed star $k$-dominating family on $G$ is the signed star $k$-domatic number of $G$, denoted by $d_{kSS}(G)$

The signed (k,k) -domatic number of digraphs

et $D$ be a finite and simple digraph with vertex set $V(D)$, and let $f:V(D)rightarrow{-1,1}$ be a two-valued function. If $kge 1$ is an integer and $sum_{xin N^-[v]}f(x)ge k$ for each $vin V(D)$, where $N^-[v]$ consists of $v$ and all vertices of $D$ from which arcs go into $v$, then $f$ is a signed $k$-dominating function on $D$. A set ${f_1,f_2,ldots,f_d}$ of distinct signed $k$-dominating functions on $D$ with the property that $sum_{i=1}^df_i(x)le k$ for each $xin V(D)$, is called a signed $(k,k)$-dominating family (of functions) on $D$. The maximum number of functions in a signed $(k,k)$-dominating family on $D$ is the signed $(k,k)$-domatic number on $D$, denoted by $d_{S}^{k}(D)$. In this paper, we initiate the study of the signed $(k,k)$-domatic number of digraphs, and we present different bounds on $d_{S}^{k}(D)$. Some of our results are extensions of well-known properties of the signed domatic number $d_S(D)=d_{S}^{1}(D)$ of digraphs $D$ as well as the signed $(k,k)$-domatic number $d_S^k(G)$ of graphs $G$

Total Italian domatic number of graphs

Let $G$ be a graph with vertex set $V(G)$. An \textit{Italian dominating function} (IDF) on a graph $G$ is a function $f:V(G)\longrightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to a vertex $u$ with $f(u)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. An IDF $f$ is called a \textit{total Italian dominating function} if every vertex $v$ with $f(v)\ge 1$ is adjacent to a vertex $u$ with $f(u)\ge 1$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct total Italian dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2$ for each vertex $v\in V(G)$, is called a \textit{total Italian dominating family} (of functions) on $G$. The maximum number of functions in a total Italian dominating family on $G$ is the \textit{total Italian domatic number} of $G$, denoted by $d_{tI}(G)$. In this paper, we initiate the study of the total Italian domatic number and present different sharp bounds on $d_{tI}(G)$. In addition, we determine this parameter for some classes of graphs

Restrained reinforcement number in graphs

A set $S$ of vertices is a restrained dominating set of a graph $G=(V,E)$ if every vertex in $V\setminus S$ has a neighbor in $S$ and a neighbor in $V\setminus S$. The minimum cardinality of a restrained dominating set is the restrained domination number $\gamma_{r}(G)$. In this paper we initiate the study of the restrained reinforcement number $r_{r}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges $F\subseteq E(\overline{G})$ for which $\gamma _{r}(G+F

RESTRAINED ROMAN REINFORCEMENT NUMBER IN GRAPHS

A restrained Roman dominating function (RRD-function) on a graph $G=(V,E)$ is a function $f$ from $V$ into $\{0,1,2\}$ satisfying: (i)  every vertex $u$ with $f(u)=0$ is adjacent to a vertex $v$ with $f(v)=2$; (ii) the subgraph induced by the vertices assigned 0 under $f$ has no isolated vertices. The weight of an RRD-function is the sum of its function value over the whole set of vertices, and the restrained Roman domination number is the minimum weight of an RRD-function on $G.$ In this paper, we begin the study of the restrained Roman reinforcement number $r_{rR}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges that we must add to the graph to decrease its restrained Roman domination number. We first show that the decision problem associated with the restrained Roman reinforcement problem is NP-hard. Then several properties as well as some sharp bounds of the restrained Roman reinforcement number are presented. In particular it is established that $r_{rR}(T)=1$ for every tree $T$ of order at least three

Signed star (k, k)-domatic number of a graph

Tyt. z nagłówka.Bibliogr. s. 619.Let G be a simple graph without isolated vertices with vertex set V(G) and edge set E(G) and let k be a positive integer. A function ƒ: E(G) →{−1, 1} is said to be a signed star k-dominating function on [formula] for every vertex v of G, where E(v) = {uv ∈ E(G) œ u ∈ N(v)}. A set {f1, f2, . . . , fd} of signed star k-dominating functions on G with the property that [formula] for each e ∈ E(G) is called a signed star (k, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (k, k)-dominating family on G is the signed star (k, k)-domatic number of G, denoted by [formula]. In this paper we study properties of the signed star (k, k)-domatic number [formula]. In particular, we present bounds on [formula], and we determine the signed (k, k)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213–218] for the signed star domatic number.Dostępny również w formie drukowanej.KEYWORDS: signed star (k, k)-domatic number, signed star domatic number, signed star k-dominating function, signed star dominating function, signed star k-domination number, signed star domination number, regular graphs