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A Reduction System for Optimal 1-Planar Graphs
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<span title="2016-10-27">2016</span>
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arXiv
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<span class="release-stage" >pre-print</span>
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<span title="2016-02-20">2016</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1602.06407v1">arXiv:1602.06407v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iiihvzvvuvdu5boovbm3eh5cqq">fatcat:iiihvzvvuvdu5boovbm3eh5cqq</a> </span>
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There is a graph reduction system so that every optimal 1-planar graph can be reduced to an irreducible extended wheel graph, provided the reductions are applied such that the given graph class is preserved. A graph is optimal 1-planar if it can be drawn in the plane with at most one crossing per edge and is optimal if it has the maximum of 4n-8 edges. We show that the reduction system is context-sensitive so that the preservation of the graph class can be granted by local conditions which can
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<a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1602.06407v2">arXiv:1602.06407v2</a>
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... e tested in constant time. Every optimal 1-planar graph G can be reduced to every extended wheel graph whose size is in a range from the (second) smallest one to some upper bound that depends on G. There is a reduction to the smallest extended wheel graph if G is not 5-connected, but not conversely. The reduction system has side effects and is non-deterministic and non-confluent. Nevertheless, reductions can be computed in linear time.
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