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Outerplanar thrackles. (English) Zbl 1234.05168
Summary: We show that a graph drawing is an outerplanar thrackle if and only if, up to an inversion in the plane, it is Reidemeister equivalent to an odd musquash. This establishes Conway’s thrackle conjecture for outerplanar thrackles. We also extend this result in two directions. First, we show that no pair of vertices of an outerplanar thrackle can be joined by an edge in such a way that the resulting graph drawing is a thrackle. Secondly, we introduce the notion of crossing rank; drawings with crossing rank 0 are generalizations of outerplanar drawings. We show that all thrackles of crossing rank 0 are outerplanar. We also introduce the notion of an \(alternating\) cycle drawing, and we show that a thrackled cycle is alternating if and only if it is outerplanar.

MSC:
05C62 Graph representations (geometric and intersection representations, etc.)
05C10 Planar graphs; geometric and topological aspects of graph theory
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