Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

In this paper, we show a polynomial-time algorithm for testing [Formula: see text]-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face. Our result is based on a reduction to the planar set of spanning trees in topological multigraphs (pssttm) problem, which is defined as follows. Given a (non-planar) topological multigraph [Formula: see text] with [Formula: see text] connected components [Formula: see text], do spanning trees of [Formula: see text] exist such that no two edges in any two spanning trees cross? Kratochvíl et al. [SIAM Journal on Discrete Mathematics, 4(2): 223–244, 1991] proved that the problem is NP-hard even if [Formula: see text]; on the other hand, Di Battista and Frati presented a linear-time algorithm to solve the pssttm problem for the case in which [Formula: see text] is a [Formula: see text]-planar topological multigraph [Journal of Graph Algorithms and Applications, 13(3): 349–378, 2009]. For any embedded flat clustered graph [Formula: see text], an instance [Formula: see text] of the pssttm problem can be constructed in polynomial time such that [Formula: see text] is [Formula: see text]-planar if and only if [Formula: see text] admits a solution. We show that, if [Formula: see text] has at most two vertices per cluster on each face, then it can be tested in polynomial time whether the corresponding instance [Formula: see text] of the pssttm problem is positive or negative. Our strategy for solving the pssttm problem on [Formula: see text] is to repeatedly perform a sequence of tests, which might let us conclude that [Formula: see text] is a negative instance, and simplifications, which might let us simplify [Formula: see text] by removing or contracting some edges. Most of these tests and simplifications are performed “locally”, by looking at the crossings involving a single edge or face of a connected component [Formula: see text] of [Formula: see text]; however, some tests and simplifications have to consider certain global structures in [Formula: see text], which we call [Formula: see text]-donuts. If no test concludes that [Formula: see text] is a negative instance of the pssttm problem, then the simplifications eventually transform [Formula: see text] into an equivalent [Formula: see text]-planar topological multigraph on which we can apply the cited linear-time algorithm by Di Battista and Frati.