Minimizing Crossings in Constrained Two-Sided Circular Graph Layouts

Circular layouts are a popular graph drawing style, where 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 that draw some 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 vertices placed on a line (the spine) 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 drawn edges for $k \ge 1$, extending the previously studied case $k=0$. We show that this relates to 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 algorithm for any fixed $k$. For the practically interesting case $k=1$ we implemented our algorithm and present experimental results that confirm the applicability of our algorithm.