2 research outputs found

    Simple realizability of complete abstract topological graphs simplified

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    An abstract topological graph (briefly an AT-graph) is a pair A=(G,X)A=(G,\mathcal{X}) where G=(V,E)G=(V,E) is a graph and X⊆(E2)\mathcal{X}\subseteq {E \choose 2} is a set of pairs of its edges. The AT-graph AA is simply realizable if GG can be drawn in the plane so that each pair of edges from X\mathcal{X} crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent Z2\mathbb{Z}_2-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.Comment: 26 pages, 17 figures; major revision; original Section 5 removed and will be included in another pape
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