Roman domination number of Generalized Petersen Graphs P(n,2)

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 $\gamma_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $\gamma_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n \geq 5)$.Comment: 9 page

k-Tuple_Total_Domination_in_Inflated_Graphs

The inflated graph $G_{I}$ of a graph $G$ with $n(G)$ vertices is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $u\in X_{i}$, $v\in X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. For integer $k\geq 1$, the $k$-tuple total domination number $\gamma_{\times k,t}(G)$ of $G$ is the minimum cardinality of a $k$-tuple total dominating set of $G$, which is a set of vertices in $G$ such that every vertex of $G$ is adjacent to at least $k$ vertices in it. For existing this number, must the minimum degree of $G$ is at least $k$. Here, we study the $k$-tuple total domination number in inflated graphs when $k\geq 2$. First we prove that $n(G)k\leq \gamma_{\times k,t}(G_{I})\leq n(G)(k+1)-1$, and then we characterize graphs $G$ that the $k$-tuple total domination number number of $G_I$ is $n(G)k$ or $n(G)k+1$. Then we find bounds for this number in the inflated graph $G_I$, when $G$ has a cut-edge $e$ or cut-vertex $v$, in terms on the $k$-tuple total domination number of the inflated graphs of the components of $G-e$ or $v$-components of $G-v$, respectively. Finally, we calculate this number in the inflated graphs that have obtained by some of the known graphs

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs