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 $f:vrightarrow{0,1,2}$ such that each vertex $vin v$ with $f(v)=0$ is adjacent to at least one vertex $uin v$ with $f(u)=2$. a total roman dominating function (trdf) of $g$ is a function $f:vrightarrow{0,1,2}$ 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, $f(v)=sum_{uin v} f(u)$, called the total roman domination number ($gamma_{tr}(g)$). similarly, a subset $ssubseteq v$ 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 $10.5operatorname{-}$factor approximation algorithm for trds problem in udgs. the running time of the proposed 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 $o(|v|+|e|)$.