On Pairs of Disjoint Segments in Convex Position in The Plane

Publisher Summary Graphs ( V , E ) are presented in this chapter, where V is a finite point set in ℝ 2 in general position and E is a collection of nondegenerate closed straight line segments with endpoints in V . Such graphs are called “geometric graphs” ( g -graphs). Denote by D a configuration comprising two disjoint segments and by CD a D configuration whose segments are sides of a convex quadrilateral. A g -graph is D -free ( CD -free) if it has no subgraph of type D ( CD ). A g -graph G = ( V , E ) is maximal D -free (maximal CD -free) if it is D -free ( CD -free) and has the maximum possible number of edges T ( n, D ) ( T ( n, CD )). The chapter describes an operation that transforms a g -graph with n vertices and e edges into one with n + 1 vertices and e + 2 edges.