A linear time algorithm to recognize clustered planar graphs and its parallelization
We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices such that all clusters induce connected subgraphs, determine whether G may be embedded into the plane such that no cluster has a hole.
This is an improvement to the O(n 2)-algorithm of Q.W. Feng et al.  and the algorithm of Lengauer  that operates in linear time on a replacement system. The size of the input of Lengauer's algorithm is not necessarily linear with respect to the number of vertices.
KeywordsPlanar Graph Maximum Weight Dual Graph Connected Subgraph Linear Time Algorithm
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