Integer linear programming models for the weighted total domination problem

A total dominating set of a graph G = (V, E) is a subset D of V such that every vertex in V (including the vertices from D) has at least one neighbour in D. Suppose that every vertex v is an element of V has an integer weight w(v) >= 0 and every edge e is an element of E has an integer weight w(e) >= 0. Then the weighted total domination (WTD) problem is to find a total dominating set D which minimizes the cost f (D) := Sigma(u is an element of D )w(u) + Sigma(e is an element of E[D]) w(e) + Sigma(v is an element of V\D) min{w(uv) vertical bar u is an element of N(v) boolean AND D}. In this paper, we put forward three integer linear programming (ILP) models with a polynomial number of constraints, and present some numerical results implemented on random graphs for WTD problem. (C) 2019 Elsevier Inc. All rights reserved.