Characterization of Graphs Based on Number of Bends in Corresponding Floor plans

Abstract

Let G = (V, E) be a maximal planar graph (MPG), where every face is triangular. A floor plan (FP) of an n-vertex MPG G is a partition of a rectangle into n rectilinear polygons called modules where two modules are adjacent if and only if there is an edge between the corresponding vertices in G. It can be easily found that it is not possible to construct a FP for a given MPG while maintaining the rectangularity of the modules of a FP (for an example, consider the complete graph K4). Hence, to satisfy adjacency requirements of a MPG, bends need to be introduced within the FPs, where a bend is a concave corner of a module in a FP. A FP with rectilinear modules or with at least one bend is called orthogonal floor plan (OFP). There exist algorithms for the construction of an OFP for a given MPG but the notion of minimum bends within an OFP is not yet discussed in the literature. In this paper, a mathematical procedure for computing the minimum number of bends required in an OFP for a MPG G has been presented. Further, it has been shown that the number of bends in an OFP depends only on critical separating triangles and K4’s.