Let $P$ be a simple polygon, $V$ its set of vertices. A minimal vertex cover $C$ of $P$ is a minimal subset of $V$ which covers $V$. The extended cover of $P$ given $C$ is the maximal subset of the boundary of $P$ covered by $C$. Let $\epsilon P$ denotes the extended cover of $P$ given $C$, and $\bar{\epsilon}P$ the complement of $\epsilon P$ with respect to $\delta P$. Denote by $\mu$ the cardinality of $\bar{\epsilon}P$. In this paper we establish lower and upper bounds on $\mu$ as a function of $n$ the cardinality of the edge set of $P$ and $k$ the cardinality of the covering set. In the restricted case where $k = 2$ we prove the bounds to be tight.