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

TL;DR

This paper proves that every graph admits an edge-outer embedding and shows that finding the shortest such embedding is NP-hard, highlighting a new complexity aspect in graph embedding problems.

Contribution

It provides a short algorithmic proof of the existence of edge-outer embeddings and establishes NP-hardness for the shortest embedding problem in certain graphs.

Findings

Every graph admits an edge-outer embedding.

Finding the shortest edge-outer embedding is NP-hard.

Introduces new problems in graph embedding studies.

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.