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We devise an algorithm for maintaining the visibility polygon of any querypoint in a dynamic polygonal domain, i.e., as the polygonal domain is modifiedwith vertex insertions and deletions to its obstacles, we update the datastructures that store the visibility polygon of the query point. Afterpreprocessing the initial input polygonal domain to build a few datastructures, our algorithm takes O(k(\lg{|VP_{\cal P'}(q)|})+(\lg{n'})^{2}+h)(resp. O(k(\lg n')^2+(\lg|VP_{\cal P'}(q)|)+h)) worst-case time to update datastructures that store visibility polygon VP_{\cal P'}(q) of a query point qwhen any vertex v is inserted to (resp. deleted from) any obstacle of thecurrent polygonal domain \cal P'. Here, n' is the number of vertices in \calP', h is the number of obstacles in \cal P', VP_{\cal P'}(q) is the visibilitypolygon of q in \cal P' (|VP_{\cal P'}(q)| is the number of vertices ofVP_{\cal P'}(q)), and k is the number of combinatorial changes in VP_{\calP'}(q) due to the insertion (resp. deletion) of v.  As an application of the above algorithm, we also devise an algorithm formaintaining the visibility graph of a dynamic polygonal domain, i.e., as thepolygonal domain is modified with vertex insertions and deletions to itsobstacles, we update data structures that store the visibility graph of thepolygonal domain. After preprocessing the initial input polygonal domain, ourdynamic algorithm takes O(k(\lg{n'})^{2}+h) (resp. O(k(\lg{n'})^{2}+h))worst-case time to update data structures that store the visibility graph whenany vertex v is inserted to (resp. deleted from) any obstacle of the currentpolygonal domain \cal P'. Here, n' is the number of vertices in \cal P', h isthe number of obstacles in \cal P', and k is the number of combinatorialchanges in the visibility graph of \cal P' due to the insertion (resp.deletion) of v.

Visibility Polygons and Visibility Graphs among Dynamic Polygonal Obstacles in the Plane