Constrained point set embedding with some given subgraphs

In this paper we study Constrained Point Set Embedding (CPSE) of graphs with a given subgraph. Let G be a planar graph, G′ be a subgraph of G and S be a set of points in general position in the plane. The aim is to find a planar drawing of G where each vertex of G is mapped to a distinct point of S, the edges of G′ are drawn as straight line segments and the number of bends in other edges is small. In this paper we prove that if G′ is an outer path, and S is a set of points in general position, then there exists a CPSE of G where all the edges of G′ are drawn as straight line segments and every other edge has at most 10 bends. More over, if G′ consists of the boundary of two faces which have a common path, then there exists a CPSE of G such that the total number of bends in G′ is at most 2 and every other edge has at most 8 bends.