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 core stability and the three
related properties can be determined efficiently.