An abstract topological graph (briefly an AT-graph ) is a pair $$A=(G,{\mathcal {X}})$$ A = ( G , X ) where $$G=(V,E)$$ G = ( V , E ) is a graph and $${\mathcal {X}}\subseteq {E \atopwithdelims ()2}$$ X ⊆ E 2 is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from $${\mathcal {X}}$$ 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 $${\mathbb {Z}}_2$$ Z 2 -realizability, where only the parity of the number of crossings for each pair of independent edges is specified.

中文翻译：

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简化的完全抽象拓扑图的简单可实现性

抽象拓扑图（简称 AT-graph）是一对 $$A=(G,{\mathcal {X}})$$ A = ( G , X ) 其中 $$G=(V,E)$$ G = ( V , E ) 是一个图，$${\mathcal {X}}\subseteq {E \atopwithdelims ()2}$$ X ⊆ E 2 是它的边对的集合。如果 G 可以在平面中绘制，使得来自 $${\mathcal {X}}$$ X 的每对边恰好相交一次并且没有其他对相交，则 AT 图 A 是可简单实现的。我们展示了简单可实现的完整 AT-图的特征在于一组有限的禁止 AT-子图，每个子图最多有六个顶点。这意味着一个简单的多项式算法，用于测试完整 AT 图形的简单可实现性，这简化了作者之前的算法。我们还展示了独立 $${\mathbb {Z}}}_2$$ Z 2 -realizability 的类似结果，