Clustered Planarity: Small Clusters in Cycles and Eulerian Graphs

We present several polynomial-time algorithms for c-planarity testing for cluster hierarchy C containing clusters of size at most three. The main result is an O(|C| + n)-time algorithm for clusters of size at most three on a cycle. The result is then generalized to a special class of Eulerian graphs, namely graphs obtained from a 3-connected planar graph of fixed size k by multiplying and then subdividing edges. An O(3 · k · n)-time algorithm is presented. We further give an O(|C| + n)-time algorithm for general 3-connected planar graphs. Submitted: December 2007 Reviewed: November 2008 Revised: February 2009 Accepted: August 2009 Final: September 2009 Published: November 2009 Article type: Regular paper Communicated by: S.-H. Hong and T. Nishizeki An extended abstract of this paper appeared in proceedings of Graph Drawing 2007. [14] Department of Applied Mathematics is supported by project 1M0021620838 of the Czech Ministry of Education. Institute for Theoretical Computer Science is supported by grant 1M0545 of the Czech Ministry of Education. The 4th author was supported by the grant GAUK 154907. The 5th author and the 6th author were supported by grant 201/05/H014 of the Czech Science Foundation. The 5th author was also partially supported by the ERASMUS program and by the DFG, project NI 369/4 (PIAF), while visiting Friedrich-Schiller-Universität Jena, Germany (October 2008–