  Let $p$ be a real number greater than one and let $\Gamma$ be a connected
graph of bounded degree. We show that the $p$-Royden boundary of $\Gamma$ with
the $p$-harmonic boundary removed is a $F_{\sigma}$-set. We also characterize
the $p$-harmonic boundary of $\Gamma$ in terms of the intersection of the
extreme points of a certain subset of one-sided infinite paths in $\Gamma$.
