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 >= 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

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Relations among Shakespeare's characters: an analysis in terms of centrality measures and new tecniques from graph theory

This work analyzes some aspects of two problems in graph theory: centrality measures that allow us to detect the most important group of nodes in a network and the clustering of a graph in coherent sub-communities. We propose two new centrality measures that are the results of a new point of view and we suggest a new algorithm to detect communities. We apply all the results to analyze drama, in particular five Shakespeare's plays