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      "statement": "# Hopcroft and Tarjan: deciding whether a network can lie in the plane\n\nThe1974publication organizes a planarity decision method around depth-first search,lowpoint-ordered paths and stack-managed placement constraints, with a linear time/space analysis in its stated adjacency-list random-access model.\n\nA network is planar if its edges can be drawn without crossings. Trying many drawings is expensive. Hopcroft and Tarjan organize the graph into paths, then use compact constraints to track which side of a cycle each piece can occupy. This lets the decision procedure avoid redoing the same placement work. The published decision test, its earlier experimental implementation and the later detailed drawing-construction algorithms are distinct parts of the history.\n\n## Scientific scope\n\nThe original algorithm tests finite simple undirected graphs using adjacency lists, a bounded-word access model and biconnected-component reduction. Linear complexity does not mean reading a dense matrix in linear vertex time. The original test does not itself output a complete drawing/embedding. A bibliographically identified erratum has not been fully assessed, and pseudocode/transcription is not offered as certified runnable code.\n\n## People and systems\n\nJohn Hopcroft; Robert Tarjan\n\n\n\n## Original sources\n\n- [Hopcroft and Tarjan:Efficient Planarity Testing(1974), original549–568](https://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/Efficient%20Planarity.pdf)\n- [ACM:Efficient Planarity Testing publication metadata](https://doi.org/10.1145/321850.321852)\n- [Mehlhorn/Mutzel:later detailed embedding phase, selected introduction only](https://kups.ub.uni-koeln.de/54688/1/zpr94-162.pdf)\n",
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