In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$ such that each vertex labelled with $0$ is adjacent to at least one vertex labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an Italian dominating function (IDF) of $G$. The weight of an IDF $f$ is $w(f)=\sum_{v\in V}f(v)$. The Italian domination number of $G$ is $γ_{I}(G)=\min_{f} w(f)$. Gao et al. (2019) have determined the value of $γ_I(P(n,3))$. In this article, we focus on the study of the Italian domination number of generalized Petersen graphs $P(n, k)$, $k\neq3$. We determine the values of $γ_I(P(n, 1))$, $γ_I(P(n, 2))$ and $γ_I(P(n, k))$ for $k\ge4$, $k\equiv2,3(\bmod5)$ and $n\equiv0(\bmod5)$. For other $P(n,k)$, we present a bound of $γ_I(P(n, k))$. With the obtained results, we partially solve the open problem presented by Brešar et al. (2007) by giving $P(n,1)$ is an example for which $γ_I=γ_{r2}$ and characterizing $P(n,2)$ for which $γ_I(P(n,2))=γ_{r2}(P(n,2))$. Moreover, our results imply $P(n,1)$ $(n\equiv0(\bmod\ 4))$ is Italian, $P(n,1)$ $(n\not\equiv0(\bmod\ 4))$ and $P(n,2)$ are not Italian.
A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the $Roman\ domination\ number$ of $G$, denoted by $γ_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $γ_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n …
An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ γ_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f…
In this article, we obtain closed formulae for the Italian domination number of rooted product graphs. As a particular case of the study, we derive the corresponding formulas for corona graphs, and we provide an alternative proof that the problem of computing the Italian domination number of a graph is NP-hard.
Let $G=(V(G),E(G))$ be a simple connected and undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $S \subseteq V(G)$ is a $dominating$ $set$ if for each $v \in V(G)$ either $v \in S$ or $v$ is adjacent to some $w \in S$. That is, $S$ is a dominating set if and only if $N[S]=V(G)$. The domination number $γ(G)$ is the minimum cardinalities of minimal dominating sets. In this paper, we give an improved upper bound on the domination number of generalized Petersen graphs $P(ck,k)$ for $c\geq 3$ and $k\geq 3$. We also prove that $γ(P(4k,k))=2k+1$ for even $k$, $γ(P(5k,k))=3k$ for all …
An Italian dominating function on a (di)graph $G$ with vertex set $V(G)$ is a function $f: V(G) \to \{0, 1, 2\}$ such that every vertex $v \in V(G)$ such that $f(v) = 0$ has an (in)neighbour assigned 2 or two (in)neighbours assigned 1. We complete the investigation of the Italian domination numbers of Cartesian products of directed cycles.
An Italian dominating function (IDF), of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V(G) $ with $ f(v) = 0, \sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V(G)}f(v) $ and Italian domination number, $ γ_I(G) $ is the minimum weight of an IDF. In this paper, we study the impact of corona operator and addition of twins on Italian domination number.
In this paper, an upper bound for the perfect Italian domination number of the cartesian product of any two graphs is obtained and the exact value of this parameter for cartesian product of some special graphs are obtained. We have also proved that for any two positive integers $a$, $b$ there exists a graph $G$ and an induced subgraph $H$ of $G$ such that $γ_I^p(G) = a$ and $γ_I^p(H) = b$. Relationship of the perfect Italian domination number with the Roman domination number and the perfect domination number of a graph $G$ are obtained and the corresponding realization problems are also solved…
A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by $γ_{[1,2]}(G)$, is called the $[1,2]$-domination number of $G$. In this paper the $[1,2]$-domination and the $[1,2]$-total domination numbers of the generalized Petersen graphs $P(n,2)$ are determined.
In this paper, we investigate the domination number of generalized Petersen graphs P(n, 2) when there is a faulty vertex. Denote by $γ(P(n,2))$ the domination number of P(n,2) and $γ(P_f(n,2))$ the domination number of P(n,2) with a faulty vertex $u_f$. We show that $γ(P_f(n,2))=γ(P(n,2))-1$ when $n=5k+1$ or $5k+2$ and $γ(P_f(n,2))=γ(P(n,2))$ for the other cases.
In a graph $G$, a vertex dominates itself and its neighbors. A subset $D \subseteq V(G)$ is a double dominating set of $G$ if $D$ dominates every vertex of $G$ at least twice. A signed graph $Σ= (G,σ)$ is a graph $G$ together with an assignment $σ$ of positive or negative signs to all its edges. A cycle in a signed graph is positive if the product of its edge signs is positive. A signed graph is balanced if all its cycles are positive. A subset $D \subseteq V(Σ)$ is a double dominating set of $Σ$ if it satisfies the following conditions: (i) $D$ is a double dominating set of $G$, and (ii) $Σ[D…
Given a graph $G = (V, E)$, a set $S \subseteq V \cup E$ of vertices and edges is called a mixed dominating set if every vertex and edge that is not included in $S$ happens to be adjacent or incident to a member of $S$. The mixed domination number $γ_{md}(G)$ of the graph is the size of the smallest mixed dominating set of $G$. We present an explicit method for constructing optimal mixed dominating sets in Petersen graphs $P(n, k)$ for $k \in \{1, 2\}$. Our method also provides a new upper bound for other Petersen graphs.
In this paper, we show that the Italian domination number of every lexicographic product graph $G\circ H$ can be expressed in terms of five different domination parameters of $G$. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let $N(v)$ denote the open neighbourhood of $v\in V(G)$, and let $w=(w_0,w_1, \dots,w_l)$ be a vector of nonnegative integers such that $ w_0\ge 1$. We say that a function $f: V(G)\longrightarrow \{0,1,\dots ,l\}$ is a $w$-dominating function if $f…
Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted by $γ_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. A function $f : V \rightarrow \{0, 1, 2\}$ is a \textit{restrained Italian dominating function} on $G$ if (i) for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N_G(v)} f(u) \geq 2$, (ii) the subgraph induced by $\{v \in V \mid f(v)=0 \}$ has no isolated vert…
Given a graph $G$ and a subset of vertices $D\subseteq V(G)$, the external neighbourhood of $D$ is defined as $N_e(D)=\{u\in V(G)\setminus D:\, N(u)\cap D\ne \varnothing\}$, where $N(u)$ denotes the open neighbourhood of $u$. Now, given a subset $D\subseteq V(G)$ and a vertex $v\in D$, the external private neighbourhood of $v$ with respect to $D$ is defined to be $epn(v,D)=\{u\in V(G)\setminus D: N(u)\cap D=\{v\}\}.$ The strong differential of a set $D\subseteq V(G)$ is defined as $\partial_s(D)=|N_e(D)|-|D_w|,$ where $D_w=\{v\in D: epn(v,D)\neq \varnothing\}$. In this paper we focus on the st…
An Italian dominating function on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. In this paper, we determine the exact values of the Italian domination numbers of some products of directed cycles.
For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$ or $v$ is adjacent to least two vertices assigned $1$ under $f$. The weight of an ID function is $\sum_{v\in V(G)}f(v)$. The Italian domination number is the minimum weight taken over all ID functions of $G$. In this paper, we initiate the study of a variant of ID functions. A restrained Italian dominating function (RID function) $f$ of $G$ is an ID function of $G$ for w…
For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ such that for each vertex $v\in V(G)$ with $f(v)=0$, $f(N(v))\geq2$, that is, either there is a vertex $u \in N(v)$ with $f (u) = 2$ or there are two vertices $x,y\in N(v)$ with $f(x)=f(y)=1$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a covering Italian dominating function (CID function) of $G$ if $f$ is an ID function and $\{v\in V(G)\mid f(v)\neq0\}$ is a vertex cover set. The covering Italian domination number (CID number) $γ_{cI}(G)$ is the minimum weight taken ove…
Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.
Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by γpr(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that γpr(G) {\leq} 4n/7 holds for every graph G with δ(G) {\geq} 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.
Given a graph $G=(V,E)$, $f:V \rightarrow \{0,1,2 \}$ is the Italian dominating function of $G$ if $f$ satisfies $\sum_{u \in N(v)}f(u) \geq 2$ when $f(v)=0$. Denote $w(f)=\sum_{v \in V}f(v)$ as the weight of $f$. Let $V_i=\{v:f(v)=i\},i=0,1,2$, we call $f$ the independent Italian dominating function if $V_1 \cup V_2$ is an independent set. The independent Italian domination number of $G$ is the minimum weight of independent Italian dominating function $f$, denoted by $i_{I}(G)$. We equivalently transform the independent domination problem of the connected block graph $G$ to the induced indepe…
Let $γ_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $γ_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if ${\rm diam}(G) = 2$ and $n(G) \le 10$, then $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ if and only if $G$ is one of seven sporadic graphs with $n(G)\le 6$ or the Petersen graph, and there are exactly ten graphs of diameter $2$ and order $11$ that attain the bound.
A set $D \subseteq V(G)$ is a \emph{dominating set} of $G$ if every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set of $G$ of minimum cardinality is called a $γ(G)$-set. For each vertex $v \in V(G)$, we define the \emph{domination value} of $v$ to be the number of $γ(G)$-sets to which $v$ belongs. In this paper, we study some basic properties of the domination value function, thus initiating \emph{a local study of domination} in graphs. Further, we characterize domination value for the Petersen graph, complete $n$-partite graphs, cycles, and paths.
A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u \in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the neighbors of $v$ is exactly two. The weight of a perfect Italian function is the sum of the weights of the vertices. The perfect Italian domination number of $G$, denoted by $γ^p_I(G)$, is the minimum weight of any perfect Italian dominating function of $G$. While introducing the parameter, Haynes and Henning (Discrete Appl. Math. (2019), 164--177) also proposed the…
An \textit{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. The \textit{weight} of an Italian dominating function $f$ is the value $ω(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$. The \textit{Italian domination number} of a digraph $D$, denoted by $γ_I(D)$, is the minimum taken over the weights of all Italian dominating functions on $D$. The \textit{Italian bondage number} of a …
We show that every connected induced subgraph of a graph $G$ is dominated by an induced connected split graph if and only if $G$ is $\cal{C}$-free, where $\cal{C}$ is a set of six graphs which includes $P_7$ and $C_7$, and each containing an induced $P_5$. A similar characterisation is shown for the class of graphs which are dominated by induced complete split graphs. Motivated by these results, we study structural descriptions of some classes of $\cal{C}$-free graphs. In particular, we give structural descriptions for the class of ($P_7$,$C_7$,$C_4$,gem)-free graphs and for the class of ($P_7…
