In this paper, we propose two new problem-specific heuristics for the dominating tree problem (DTP), a variant of the well-known minimum dominating set problem. Given a connected, undirected, and edge-weighted graph \(G = (V, E)\) , where V denotes the set of vertices and E denotes the set of edges, with each edge \(e \in E\) associated with a positive weight, the objective of the DTP is to construct a tree \(T \subseteq G\) with minimum sum of edge-weights such that every vertex \(v \in V\) is either included in T or is adjacent to at least one vertex in T. The DTP is known to be an \(\mathcal{N}\mathcal{P}\) -hard problem and has various real-world applications, particularly in wireless sensor networks. We evaluate the performance of the proposed heuristics against existing problem-specific heuristics for the DTP using benchmark instances from the literature. The results show that our proposed heuristics outperform the existing problem-specific heuristics on most of the instances.