Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12‐sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10‐sided rectilinear polygons and runs in O(n) time. We also describe a linear‐time algorithm for proportional contact representation of planar 3‐trees with 8‐sided rectilinear polygons and show that this is optimal, as there exist planar 3‐trees that require 8‐sided polygons. We then show that a maximal outer‐planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer‐boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series‐parallel graphs. Here we show that O(1)‐sided rectilinear polygons are not possible unless we allow holes, but 6‐sided polygons can be achieved with arbitrarily small holes.