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arXiv · 1710.09048

Edge-outer graph embedding and the complexity of the DNA reporter strand problem

M. N. Ellingham·Joanna A. Ellis-Monaghan
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Abstract

In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a DNA reporter strand, that is, a closed walk covering every edge either once or twice, in opposite directions if twice, and passing through each vertex in a particular way. This corresponds to showing that every graph has an \emph{edge-outer embedding}, that is, an orientable embedding with some face that is incident with every edge. In the motivating application, the objective is such a closed walk of minimum length. Here we give a short algorithmic proof of the original existence result, and also prove that finding a shortest length solution is NP-hard, even for $3$-connected cubic ($3$-regular) planar graphs. Independent of the motivating application, this problem opens a new direction in the study of graph embeddings, and we suggest new problems emerging from it.

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M. N. Ellingham, Joanna A. Ellis-Monaghan. 2017-10-25. Edge-outer graph embedding and the complexity of the DNA reporter strand problem. https://arxiv.org/abs/1710.09048

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