Efficient Algorithms for Drawing Planar Graphs

This thesis presents efficient algorithms for drawing planar graphs. Graph drawing addresses the problem of constructing geometric representation of information and finds applications in almost every branch of science and technology. In this thesis we study rectangular drawings, orthogonal drawings, convex drawings and our newly defined box-rectangular drawings. Rectangular drawings and box-rectangular drawings have practical applications in VLSI floorplanning and in architectural floorplanning, whereas orthogonal drawings have applications in circuit schematics, data-flow diagrams etc. We give a linear-time algorithm for finding a rectangular drawing of a plane graph if it exists, and obtain the best possible upper bound on grid size for a rectangular drawing. Minimizing the number of bends in an orthogonal drawing is a challenging problem. We give a linear-time algorithm for finding an orthogonal drawing of a triconnected cubic plane graph with the minimum number of bends. We introduce a new drawing style called box-rectangular drawing and obtain a necessary and sufficient condition for a plane graph to have a box-rectangular drawing. We also present the corresponding linear-time algorithm. We newly define 4-canonical decomposition and give a linear-time algorithm for finding a 4-canonical decomposition of a 4-connected planar graph. Our 4-canonical decomposition has applications in finding a convex drawing of a 4-connected planar graph.