Characterization of perfect Roman domination edge critical‎ ‎trees

‎A perfect Roman dominating function on a graph $G =(V‎, ‎E)$ is a function $f‎: ‎V \longrightarrow \{0‎, ‎1‎, ‎2\}$‎ ‎satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex‎ ‎$v$ for which $f(v) = 2$‎. ‎The weight of a perfect Roman dominating function $f$ is the sum of‎ ‎the weights of the vertices‎. ‎The perfect Roman domination number of $G$‎, ‎denoted by $\gamma_{R}^{p}(G)$‎, ‎is‎ ‎the minimum weight of a perfect Roman dominating function in $G$‎. ‎In this paper‎, ‎we study the‎ ‎graphs for which adding any new edge decreases the perfect Roman‎ ‎domination number‎. ‎We call these graphs $\gamma_R^p$-edge critical‎. ‎The purpose of this paper is to characterize the class of‎ ‎$\gamma_R^p$-edge critical trees‎

Perfect Roman Domination and Unique Response Roman Domination

The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature