Abstract
In recent years, graph drawing plays an important role in computational geometry and graph theory. There are many di fferent ways to represent graphs, for example, straight line drawing and orthogonal drawing. In our thesis, we focus on the visibility representation (or VR for short) of planar graphs. Let G be a plane graph with n vertices. In a visibility representation of G, each vertex of G is represented as a horizontal line segment (called vertex segment), and two vertex segments are connected by a vertical line segment (called edge segment) if the corresponding vertices are adjacent in G. Rosenstiehl and Tarjan conjectured that minimizing the area of visibility representation is NP-hard. In this thesis, we reduce from 3SAT problem and show the area-minimization problem of visibility representation is NP-complete. A visibility representation with congruent segments (or VRCS for short) is a variant of visibility representation, which is similar to VR, but all vertices of G are represented as equal length horizontal line segments. We show that the VRCS drawing problem for general planar graphs is NP-complete. We also prove the problem of visibility representation with congruent segments of fixed length (or VRCSFL for short) on planar graphs is NP-complete. Apart from NP-hardness proofs, we provide a VRCS drawing algorithm for outerplane graphs in O(n) time, and an O(n) time VRCS drawing algorithm for bipartite plane graphs, respectively.