We obtain new results on 2-rainbow domination number of generalized Petersen graphs $P(5k,k)$. In some cases (for some infinite families), exact values are established, and in all other cases lower and upper bounds are given. In particular, it is shown that, for $k>3$, ${\gamma }_{r2}\left(P(5k,k)\right)=4k$ for $k\equiv 2,8mod10$, ${\gamma }_{r2}\left(P(5k,k)\right)=4k+1$ for $k\equiv 5,9mod10$, $4k+1\le {\gamma }_{r2}\left(P(5k,k)\right)\le 4k+2$ for $k\equiv 1,6,7mod10$, and $4k+1\le {\gamma }_{r2}\left(P(5k,k)\right)\le 4k+3$ for $k\equiv 0,3,4mod10$.

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs $P(6k,k)$. In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

A double Roman dominating function on a graph $G=(V,E)$ is a function $f:V\to \{0,1,2,3\}$ with the properties that if $f\left(u\right)=0$, then vertex u is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and if $f\left(u\right)=1$, then vertex u is adjacent to at least one vertex assigned 2 or 3. The weight of f equals $w\left(f\right)={\sum }_{v\in V}f\left(v\right)$. The double Roman domination number ${\gamma }_{dR}\left(G\right)$ of a graph G is the minimum weight of a double Roman dominating function of G. A graph is said to be double Roman if ${\gamma }_{dR}\left(G\right)=3\gamma \left(G\right)$, where $\gamma \left(G\right)$ is the domination number of G. We obtain the sharp lower bound of the double Roman domination number of generalized Petersen graphs $P(3k,k)$, and we construct solutions providing the upper bounds, which gives exact values of the double Roman domination number for all generalized Petersen graphs $P(3k,k)$. This implies that $P(3k,k)$ is a double Roman graph if and only if either $k\equiv 0$ (mod 3) or $k\in \{1,4\}$.

A subset P of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from P. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of G, and is denoted by ψk(G). It is known that the problem of finding a minimum 3-path vertex cover is NP-hard for planar graphs. In this paper we establish an upper bound for ψ3(G), where G is from an important family of planar graphs, called hexagonal graphs, arising from real world applications.

Mixed fault diameter of Cartesian graph bundles II