Computing β-Drawings of 2-Outerplane Graphs in Linear Time

Abstract

A straight-line drawing of a plane graph G is a drawing of G where each vertex is drawn as a point and each edge is drawn as a straight-line segment without edge crossings. A proximity drawing Γ of a plane graph G is a straight-line drawing of G with the additional geometric constraint that two vertices of G are adjacent if and only if no other vertex of G is drawn in Γ within a “proximity region” of these two vertices in Γ. Depending upon how the proximity region is defined, a given plane graph G may or may not admit a proximity drawing. In one class of proximity drawings, known as β-drawings, the proximity region is defined in terms of a parameter β, where β ∈ [0, ∞ ). A plane graph G is β-drawable if G admits a β-drawing. A sufficient condition for a biconnected 2-outerplane graph G to have a β-drawing is known. However, the known algorithm for testing the sufficient condition takes time O(n ²). In this paper, we give a linear-time algorithm to test whether a biconnected 2-outerplane graph G satisfies the known sufficient condition or not. This consequently leads to a linear algorithm for β-drawing of a wide subclass of biconnected 2-outerplane graphs.