A Linear-Time Algorithm for Symmetric Convex Drawings of Internally Triconnected Plane Graphs

Abstract

Symmetry is one of the most important aesthetic criteria in Graph Drawing which can reveal the hidden structure in the graph. Convex drawing is a straight-line drawing where every facial cycle is drawn as a convex polygon. In this paper, we prove that given an internally triconnected plane graph with symmetries, there exists a convex drawing of the graph which displays the given symmetries. We present a linear-time algorithm for constructing symmetric convex drawings of internally triconnected planar graphs. This is an extension of a classical result due to Tutte (Proc. Lond. Math. Soc. 10(3):304–320, 1960; Proc. Lond. Math. Soc. 13:743–768, 1963) who proved that every triconnected plane graph with a given convex polygon as a boundary admits a convex drawing. Note that Tutte’s barycenter mapping method can be implemented in O(n 1.5) time and O(nlog n) space at best (Lipton et al. in SIAM J. Numer. Anal. 16:346–358, 1979). Our divide and conquer algorithm explicitly exploits the fundamental properties of symmetric drawing, which consists of congruent drawings of isomorphic subgraphs. We first find an isomorphic subgraph of a given symmetric plane graph, and compute an angle-constrained convex drawing of the subgraph. Finally, a symmetric convex drawing of the given graph is constructed by merging repetitive copies of the congruent drawings of isomorphic subgraphs. For this purpose, we define a new problem of angle-constrained convex drawing of plane graphs, where some of outer vertices have angle constraints. Our results also imply that there is a linear-time algorithm that constructs maximally symmetric convex drawings of triconnected planar graphs. Previous algorithm (Hong et al. in Discrete Comput. Geom. 36:283–311, 2006) constructs symmetric drawings of triconnected planar graphs with straight-lines.