Eulerian Walks in Temporal Graphs

Abstract An Eulerian walk (or trail) is a (resp. that visits every edge of graph G at least exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges Königsberg problem in 1736. But what if had to take bus? In temporal $$\varvec{(G,\lambda )}$$ ( G , λ ) , with $$\varvec{\lambda : E(G)}\varvec{\rightarrow } \varvec{2}^{\varvec{[\tau ]}}$$ : E → 2 [ τ ] an $$\varvec{e}\varvec{\in \varvec{E(G)}$$ e ∈ available only times specified (e)}\varvec{\subseteq \varvec{[\tau ]}$$ ⊆ same way connections public transportation network city or sightseeing tours are scheduled times. this paper, we deal walks, local trails, and respectively referring traversal no constraints, constrained not repeating single timestamp, never throughout entire traversal. We show that, edges always available, then deciding whether has trail polynomial, it $$\varvec{\texttt {NP}}$$ NP -complete even $$\varvec{\tau = 2}$$ = . contrast, general case, any these problems -complete, under very strict hypotheses. finally give {XP}}$$ XP algorithms parametrized }$$ for +tw(G)}$$ + t w trails where $$\varvec{tw(G)}$$ refers treewidth $$\varvec{G}$$