Brief Announcement: The Shortest Temporal Exploration Problem

Abstract

A temporal graph is a graph for which the edge set can change from one time step to the next. This paper considers undirected temporal graphs defined over L time steps and connected at each time step. We study the Shortest Temporal Exploration Problem (STEXP) that, given the evolution of the graph, asks for a temporal walk that starts at a given vertex, moves over at most one edge at each time step, visits all the vertices, takes at most L time steps and traverses the smallest number of edges. We prove that every constantly connected temporal graph with n vertices can be explored with O(n^{1.5}) edges traversed if L is O(n^{3.5}) time steps. This result improves the upper bound of O(n²) edges when L is Ω(n²). Moreover, we study the case where the graph has a diameter bounded by a parameter k at each time step and we prove that there exists an exploration which takes O(kn²) time steps and traverses O(kn) edges. Finally, the case where the underlying graph is a cycle is studied and tight linear bounds are provided on the number of edges traversed in the worst-case.