Total Roman domination and total domination in unit disk graphs

Let G = (V, E) be a simple, undirected and connected graph. A Roman dominating function (RDF) on the graph G is a function f : V -+ {0, 1, 2} such that each vertex v E V with f (v) = 0 is adjacent to at least one vertex u E V with f (u) = 2. A total Roman dominating function (TRDF) of G is a function f : V -+ {0, 1, 2} such that (i) it is a Roman dominating function, and (ii) the vertices with non -zero weights induce a subgraph with no isolated vertex. The total Roman dominating set (TRDS) problem is to minimize the associated weight, f(V) = Eu is an element of V f (u), called the total Roman domination number (gamma tR(G)). Similarly, a subset S C V is said to be a total dominating set (TDS) on the graph G if (i) S is a dominating set of G, and (ii) the induced subgraph G[S] does not have any isolated vertex. The objective of the TDS problem is to minimize the cardinality of the TDS of a given graph. The TDS problem is NP-complete for general graphs. In this paper, we propose a simple 10.5 -factor approximation algorithm for TRDS problem in UDGs. The running time of the prop osed algorithm is O(|V | log k), where k is the number of vertices with weights 2. It is an improvement over the best-known 12-factor approximation algorithm with running time O(|V | log k) available in the literature. Next, we propose another algorithm for the TDS problem in UDGs, which improves the previously best-known approximation factor from 8 to 7.79. The running time of the proposed algorithm is O(|V | + |E|).