LATIN 2006: LATIN 2006: Theoretical Informatics pp 737-744

# Minimal Eulerian Circuit in a Labeled Digraph

• Eduardo Moreno
• Martín Matamala
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3887)

## Abstract

Let G = (V,A) be an Eulerian directed graph with an arc-labeling. In this work we study the problem of finding an Eulerian circuit of lexicographically minimal label among all Eulerian circuits of the graph. We prove that this problem is NP-hard by showing a reduction from the Directed-Hamiltonian-Circuit problem.

If the labeling of the arcs is such that arcs going out from the same vertex have different labels, the problem can be solved in polynomial time. We present an algorithm to construct the unique Eulerian circuit of lexicographically minimal label starting at a fixed vertex. Our algorithm is a recursive greedy algorithm which runs in $${\mathcal O}$$(|A|) steps.

We also show an application of this algorithm to construct the minimal De Bruijn sequence of a language.

## Keywords

Hamiltonian Circuit Eulerian Circuit Starting Vertex Minimal Label Minimal Slope
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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