On Sets of Line Segments Featuring a Cactus Structure

In this paper we derive sharp upper and lower bounds on the number of intersections and closed regions that can occur in a set of line segments whose underlying planar graph is a cactus graph. These bounds can be used to evaluate the complexity of certain algorithms for problems defined on sets of segments in terms of the cardinality of the segment sets. In particular, we give an application in the problem of finding a path between two points in a set of segments which travels through a minimum number of segments.