Upper and lower bounds on approximating weighted mixed domination

A mixed dominating set of a graph G=(V,E) is D vertices and edges, such that for every edge or vertex, if it not in D, then adjacent incident to at least one vertex D. The domination problem find with minimum cardinality. It has applications system control some other scenarios NP-hard compute an optimal solution. This paper studies approximation algorithms hardness the weighted problem. version generalization unweighted version, where all are assigned same nonnegative weight wv edges we, question total weight. Although simple 2-approximation algorithm, few results known. main contributions this include: we≥wv, algorithm; we≥2wv, inapproximability within ratio 1.3606 unless P=NP 2 under UGC; 2wv>we≥wv, 1.1803 1.5 we0.