  We devise the following dynamic algorithms for both maintaining as well as
querying for the visibility and weak visibility polygons amid vertex insertions
and/or deletions to the simple polygon. * A fully-dynamic algorithm for
maintaining the visibility polygon of a fixed point located interior to the
simple polygon amid vertex insertions and deletions to the simple polygon. The
time complexity to update the visibility polygon of a point $q$ due to the
insertion (resp. deletion) of vertex $v$ to (resp. from) the current simple
polygon is expressed in terms of the number of combinatorial changes needed to
the visibility polygon of $q$ due to the insertion (resp. deletion) of $v$. *
An output-sensitive query algorithm to answer the visibility polygon query
corresponding to any point $p$ in $\mathbb{R}^2$ amid vertex insertions and
deletions to the simple polygon. If $p$ is not exterior to the current simple
polygon, then the visibility polygon of $p$ is computed. Otherwise, our
algorithm outputs the visibility polygon corresponding to the exterior
visibility of $p$. * An incremental algorithm to maintain the weak visibility
polygon of a fixed-line segment located interior to the simple polygon amid
vertex insertions to the simple polygon. The time complexity to update the weak
visibility polygon of a line segment $pq$ due to the insertion of vertex $v$ to
the current simple polygon is expressed in terms of the sum of the number of
combinatorial updates needed to the geodesic shortest path trees rooted at $p$
and $q$ due to the insertion of $v$. * An output-sensitive algorithm to compute
the weak visibility polygon corresponding to any query line segment located
interior to the simple polygon amid both the vertex insertions and deletions to
the simple polygon. Each of these algorithms requires preprocessing the initial
simple polygon.
