In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ( @math ) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A @math -planar graph is a planar graph in which each vertex has degree less or equal to @math . In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size @math . For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge.
Given a planar graph G together with a planar representation P, a region preserving grid embedding of G is a planar embedding of G in the rectilinear grid that has planar representation isomorphic to P. In this paper, an algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges. This algorithm makes use of network flow techniques, and runs in time @math , where n is the number of vertices of the graph. Constrained versions of the problem are also considered, and most results are extended to k-gonal graphs, i.e., graphs whose edges are sequences of segments with slope multiple of @math degrees. Applications of the above results can be found in several areas: VLSI circuit layout, architectural design, communication by light or microwave, transportation problems, and automatic layout of graphlike diagrams. A directed graph is upward planar if it can be drawn in the plane such that every edge is a monotonically increasing curve in the vertical direction and no two edges cross. An undirected graph is rectilinear planar if it can be drawn in the plane such that every edge is a horizontal or vertical segment and no two edges cross. Testing upward planarity and rectilinear planarity are fundamental problems in the effective visualization of various graph and network structures. For example, upward planarity is useful for the display of order diagrams and subroutine-call graphs, while rectilinear planarity is useful for the display of circuit schematics and entity-relationship diagrams. We show that upward planarity testing and rectilinear planarity testing are NP-complete problems. We also show that it is NP-hard to approximate the minimum number of bends in a planar orthogonal drawing of an n-vertex graph with an @math error for any @math . We connect two aspects of graph drawing, namely angular resolution, and the possibility to draw with all angles an integer multiple of 2π d. A planar graph with angular resolution at least π 2c an be drawn with all angles an integer multiple of π 2 (rectilinear). For d =4 , d> 2, an angular resolution of 2π d does not imply that the graph can be drawn with all angles an integer multiple of 2π d. We argue that the exceptional situation for d = 4 is due to the absence of triangles in the rectangular grid.