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The Minimum Weighted Connected Vertex Cover problem (MWCVC) is to find a subset <inline-formula> <tex-math notation="LaTeX">$F\subset V(G)$ </tex-math></inline-formula> with minimum weight in a node-weighted graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, such that when removing the set <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>, the inducing graph of remaining vertices holds no edges, and the graph induced from set <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is required to be connected. This problem comes from the classical combinatorial problem in graph theory, i.e., the Vertex Cover Problem. A large number of results on algorithms for the MWCVC problem have been reported. In this paper, we proposed two heuristic algorithms, denoted as VCC and LCVCC, to find a connected vertex cover set in a general weighted graph. The time complexity of both two algorithms are less than <inline-formula> <tex-math notation="LaTeX">$O(n^{4})$ </tex-math></inline-formula>. We compare these two algorithms with two known heuristic algorithms GR and GD (proposed by Dagdeviren in 2021) on connected graphs, and draw a conclusion that both of VCC and LCVCC perform better than GR or GD. Relatively speaking, LCVCC is expected to have better performance in dense graphs than VCC.

Two Heuristic Algorithms for the Minimum Weighted Connected Vertex Cover Problem Under Greedy Strategy