# Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736. What if Euler had to take a bus? In a temporal graph (G,λ), with λ: E(G)→ 2^[τ], an edge e∈ E(G) is available only at the times specified by λ(e)⊆ [τ], in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this scenario, even though several translations of Eulerian trails and walks are possible in temporal terms, only a very particular variation has been exploited in the literature, specifically for infinite dynamic networks (Orlin, 1984). In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether (G,λ) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is NP-complete even if it has lifetime 2. In contrast, in the general case, solving any of these problems is NP-complete, even under very strict hypothesis.

• 6 publications
• 18 publications
research
02/16/2021

### Metropolis Walks on Dynamic Graphs

Recently, random walks on dynamic graphs have been studied because of it...
research
12/21/2019

### Complexity results for the proper disconnection of graphs

For an edge-colored graph G, a set F of edges of G is called a proper ed...
research
03/06/2013

### Deciding Morality of Graphs is NP-complete

In order to find a causal explanation for data presented in the form of ...
research
07/31/2018

### On Exploring Temporal Graphs of Small Pathwidth

We show that the Temporal Graph Exploration Problem is NP-complete, even...
research
02/10/2020

### Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players

We analyze the computational complexity of several new variants of edge-...
research
11/16/2021

### On The Complexity of Maximizing Temporal Reachability via Trip Temporalisation

We consider the problem of assigning appearing times to the edges of a d...
research
01/14/2022

### Delay-Robust Routes in Temporal Graphs

Most transportation networks are inherently temporal: Connections (e.g. ...