Geometric Embedding of Path and Cycle Graphs in Pseudo-convex Polygons

Given a graph $G$ with $n$ vertices and a set $S$ of $n$ points in the plane, a point-set embedding of $G$ on $S$ is a planar drawing such that each vertex of $G$ is mapped to a distinct point of $S$. A straight-line point-set embedding is a point-set embedding with no edge bends or curves. The point-set embeddability problem is NP-complete, even when $G$ is $2$-connected and $2$-outerplanar. It has been solved polynomially only for a few classes of planar graphs. Suppose that $S$ is the set of vertices of a simple polygon. A straight-line polygon embedding of a graph is a straight-line point-set embedding of the graph onto the vertices of the polygon with no crossing between edges of graph and the edges of polygon. In this paper, we present $O(n)$-time algorithms for polygon embedding of path and cycle graphs in simple convex polygon and same time algorithms for polygon embedding of path and cycle graphs in a large type of simple polygons where $n$ is the number of vertices of the polygon.