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In this paper, we focus on the core stability of vertex cover games, which arise from vertex cover problems on graphs. Based on duality theory of linear programming, we prove that a balanced vertex cover game has a stable core if and only if every edge belongs to a maximum matching in the underlying graph. We also prove that for a totally balanced vertex cover game, the core largeness, extendability, and exactness are all equivalent, which implies core stability. Furthermore, we show that coredoi:10.1080/15427951.2008.10129174 fatcat:sv5g6wajorcc5fourzhuwugede