Mixed Domination on Trees

‎A mixed dominating set $S$ of a graph $G(V,E)$ is a subset $ S \subseteq V \cup E$ such that each element $r\in (V \cup E) \setminus S$ is adjacent or incident to at least one element of $S$‎. ‎The mixed domination number $\gamma_m(G)$ of a graph $G$ is the minimum cardinality among all mixed dominating sets in $G$‎. ‎%The minimum cardinality of all mixed dominating sets of $G$ is denoted by $\gamma_{m}(G)$‎. ‎The problem of finding $\gamma_{m}(G)$ is an NP-complete problem‎. ‎We present a dynamic programming algorithm on tree-width of trees‎. ‎This algorithm construct a mixed dominating set of size $\gamma_{m}(G)$ on a tree in polynomial-time and space