Graph algorithms and NP-completeness
Graph algorithms and NP-completeness
A linear algorithm for embedding planar graphs using PQ-trees
Journal of Computer and System Sciences
An O(m log n)-time algorithm for the maximal planar subgraph problem
SIAM Journal on Computing
Theoretical Computer Science
Stop minding your p's and q's: a simplified O(n) planar embedding algorithm
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Journal of the ACM (JACM)
Depth-First Search and Kuratowski Subgraphs
Journal of the ACM (JACM)
A Linear Algorithm for the Maximal Planar Subgraph Problem
WADS '95 Proceedings of the 4th International Workshop on Algorithms and Data Structures
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
Journal of Computer and System Sciences
A note on computing a maximal planar subgraph using PQ-trees
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
Theoretical Computer Science
Experiments on exact crossing minimization using column generation
Journal of Experimental Algorithmics (JEA)
A branch-and-cut approach to the crossing number problem
Discrete Optimization
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Untangling graphs representing spatial relationships driven by drawing aesthetics
Proceedings of the 17th Panhellenic Conference on Informatics
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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. An important 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 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 on the creation and destruction of biconnected components in which the PC-tree plays a major role. Previously, there have been reports that the MPS problem can be solved in linear time. However, there was no concrete data structure realizing them.