Perfect triple Roman domination

Abstract

Let $f$ be a function that assigns labels from the set $\{0,1,2,3,4\}$ to the vertices of a simple graph $G$. The active neighborhood $AN\left(v\right)$ of a vertex $v\in V\left(G\right)$ with respect to $f$ is the set of all neighbors of $v$ that are assigned non-zero values under $f$. The function $f$ is a perfect triple Roman dominating function (PTRD-function) on $G$ if for every vertex $v\in V\left(G\right)$ with $f\left(v\right)<3$, we have ${\sum }_{u\in N\left[v\right]}f\left(u\right)=\left|AN\left(v\right)\right|+3$. The weight of a PTRD-function is the sum of its function values over the whole set of vertices, and the PTRD-number is the minimum weight of a PTRD-function on $G$. In this paper, we show that determining the PTRD-number is NP-complete even when restricted to bipartite graphs. Moreover, the exact values of the PTRD-number for paths and cycles are established. Moreover, we provide an upper bound for the PTRD-number for trees of order at least five and we characterize the extremal trees attaining this upper bound.