  An incompressible surface $F$ on the boundary of a compact orientable
3-manifold $M$ is arc-extendible if there is an arc $\gamma$ on $\partial M - $
Int $F$ such that $F \cup N(\gamma)$ is incompressible, where $N(\gamma)$ is a
regular neighborhood of $\gamma$ in $\partial M$. Suppose for simplicity that
$M$ is irreducible, and $F$ has no disk components. If $M$ is a product
$F\times I$, or if $\partial M - F$ is a set of annuli, then clearly $F$ is not
arc-extendible. The main theorem of this paper shows that these are the only
obstructions for $F$ to be arc-extendible.
