Minimizing Crossings In Constrained Two-Sided Circular Graph Layouts

Abstract

Circular graph layout is a popular drawing style, in which vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is NP-hard. One way to allow for fewer crossings in practice are two-sided layouts, which drawsome edges as curves in the exterior of the circle. In fact, one- and two-sided circular layouts are equivalent to one-page and two-page book drawings, i.e., graph layouts with all verticesplaced on a line (thespine) and edges drawn in one or two distinct half-planes (the pages) bounded by the spine. In this paper we study the problem of minimizing the crossings for a fixed cyclic vertex order by computing an optimal k-plane set of exteriorly drawnedges for k ≥ 1, extending the previously studied case k = 0. We show that this relatesto finding bounded-degree maximum-weight induced subgraphs of circle graphs, which is a graph-theoretic problem of independent interest. We show NP-hardness for arbitrary k, present an efficient algorithm for k = 1, and generalize it to an explicit XP-time algorithmfor any fixed k. For the practically interesting case k= 1 we implemented our algorithm and present experimental results that confirm its applicability.