### Abstract:

A total Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} such that every vertex v with f(v) = 0 is adjacent to some vertex u with f(u) = 2, and the subgraph of G induced by the set of vertices w such that f(w) > 0 has no isolated vertices. The weight of f is Σv∈V (G)f(v). The total Roman domination number γtR(G) is the minimum weight of a total Roman dominating function on G. A graph G is γtR-edge-supercritical if γtR(G + e) = γtR(G) − 2 for every e ∈ E(G) = ∅, and γtR-edge-stable if γtR(G + e) = γtR(G) for every e ∈ E(G) = ∅. For an edge e ∈ E(G) incident with a degree 1 vertex, we deﬁne γtR(G − e) = ∞. A graph G is γtR-ER-critical if γtR(G − e) > γtR(G) for every e ∈ E(G), γtR-ER-supercritical if γtR(G − e) ≥ γtR(G) + 2 for every e ∈ E(G), and γtR-ER-stable if γtR(G − e) = γtR(G) for every e ∈ E(G). We characterize γtR-ER-critical and γtR-ER-supercritical graphs. In addition, we investigate connected γtR-edge-supercritical graphs and exhibit inﬁnite classes of such graphs. We present a connection between γtR-ER-supercritical and γtR-edge-stable graphs, and similarly between γtR-edge-supercritical and γtR-ER-stable graphs.