A Linear Time Algorithm for Finding a Maximal Planar Subgraph Based on PC-Trees

In Shih & Hsu, a planarity test was introduced utilizing a data structure called PC-trees, generalized from PQ-trees. They illustrated that a PC-tree is more natural IN representing planar graphs. Their algorithm starts by constructing a depth-first-search tree and adds all back edges to a vertex one by one. A major feature in the S&H algorithm is that, at each iteration, at most two terminal nodes need to be computed and the unique tree path between these two nodes provides essentially the boundary path of the newly formed biconnected component. In this paper we modify their PC-tree algorithm and introduce the deferred planarity test (DPT), which has the added benefit of finding a maximal planar subgraph (MPS) in linear time when the given graph is not planar. DPT is an incremental algorithm, which does not determine the terminal nodes and only computes a partial terminal path at each iteration. DPT continually deletes back edges that could create a violation to the formation of those partial terminal paths so that, at the end, the subgraph constructed is guaranteed to be planar. The key to the efficiency of the S&H and the DPT algorithms lies in their management of biconnected components in which the PC-tree plays a major role. The DPT algorithm has an even more concise description than that of the S&H algorithm. Previously, there have been reports that the MPS problem can be solved in linear time. However, there was no concrete data structure realizing them.