C-planarity of Embedded Cyclic c-Graphs

TL;DR

This paper presents a quadratic-time algorithm for testing c-planarity in flat clustered graphs with three clusters, based on fixed combinatorial embedding, by reducing the problem to perfect matching in planar bipartite graphs.

Contribution

It introduces a new efficient algorithm for c-planarity testing in a specific class of cyclic clustered graphs with three clusters, expanding the understanding of graph embedding problems.

Findings

C-planarity is solvable in quadratic time for flat three-cluster graphs with fixed embedding.

Reduction of c-planarity testing to perfect matching in planar bipartite graphs.

Applicable to cyclic clustered graphs where the contracted graph forms a cycle.

Abstract

We show that c-planarity is solvable in quadratic time for flat clustered graphs with three clusters if the combinatorial embedding of the underlying graph is fixed. In simpler graph-theoretical terms our result can be viewed as follows. Given a graph $G$ with the vertex set partitioned into three parts embedded on a 2-sphere, our algorithm decides if we can augment $G$ by adding edges without creating an edge-crossing so that in the resulting spherical graph the vertices of each part induce a connected sub-graph. We proceed by a reduction to the problem of testing the existence of a perfect matching in planar bipartite graphs. We formulate our result in a slightly more general setting of cyclic clustered graphs, i.e., the simple graph obtained by contracting each cluster, where we disregard loops and multi-edges, is a cycle.