Notes on double Roman domination edge critical graphs

Given a graph G = (V, E), a double Roman dominating function (DRDF) on a graph G is a function f : V -> {0, 1, 2, 3} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 3 or two vertices v1 and v2 for which f(v(1)) = f(v(2)) = 2, and every vertex u for which f(u) = 1 is adjacent to at least one vertex v for which f(v) >= 2. The weight w (f) of a double Roman dominating function f is the value w(f) = Sigma(u is an element of V) f(u). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G, denoted by gamma(dR)(G). We say that G is gamma dR-edge critical, if gamma(dR)(G + e) < gamma(dR)(G) for each e is an element of E(<(G)over bar>), where (G) over bar is the complement of G, and k-gamma(dR)-edge supercritical if gamma(dR)(G) = k and gamma(dR)(G + e) = gamma(dR)(G) - 2 for every edge e is an element of E((G) over bar). In this paper, we characterize gamma(dR)-edge critical trees, answering a problem posed by Nazari-Moghaddam and Volkmann (Discrete Math. Algorithms App. 12 (2020) 2050020). Moreover, we investigate connected k-gamma(dR)-edge supercritical graphs for k is an element of {5, 6, 7, 8}.