Vertex-edge perfect Roman domination number

A vertex-edge perfect Roman dominating function on a graph $G = (V, E)$ (denoted by ve-PRDF) is a function $f:V\left(G\right)\longrightarrow\{0, 1, 2\}$ such that for every edge $uv\in E$, $\max\{f(u), f(v)\}\neq0$, or $u$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$, or $v$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$. The weight of a ve-PRDF on $G$ is the sum $w(f) = \sum_{v\in V}f(v)$. The vertex-edge perfect Roman domination number of $G$ (denoted by $\gamma_{veR}^{p}(G)$) is the minimum weight of a ve-PRDF on $G$. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $T$, we give upper and lower bounds for $\gamma_{veR}^{p}(T)$ in terms of the order $n$, $l$ leaves and $s$ support vertices. Lastly, we determine $\gamma_{veR}^{p}(G)$ for Petersen, cycle and Flower snark graphs

Total Perfect Roman Domination

A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑v∈Vf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)≤3γp(G) for every graph G, and we characterized trees T of order n≥3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties