The Shortest Temporal Exploration Problem

TL;DR

It is proved that every constantly connected temporal graph with n vertices can be explored with O(n 1.5) edges traversed within O(n 3.5) time steps and the upper bound of O(n 2) edges for an exploration is improved by the upper bound of time steps for an exploration which is also O(n 2).

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 all 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 within O(n 3.5 ) time steps. This result improves the upper bound of O(n 2 ) edges for an exploration provided by the upper bound of time steps for an exploration which is also O(n 2 ). Morever, 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(kn 2 ) time steps and traverses O(kn) edges. Finally, the case where the underlying graph is a cycle is studied and tight bounds are provided on the number of edges traversed in the worst-case if L $\ge$ 2n -3.

Figures (5)

• Figure 1: A representation of a temporal graph with $L=3$. Two examples of $X_2$ sets are presented: the subsets of vertices colored blue ($A$ and $D$) and green ($E$ and $C$).
• Figure 2: Representation of a subset of vertices $S$ in the graph $\mathcal{G}_{[t_1,t_2]}$ with $t_1+4\frac{n^{2.5}}{\sqrt{q}}\leq t_2$. We represent in blue the vertices of $X_k^S$, with $|X_k^S|<\sqrt{n/q}$, and in green the vertices of $S\backslash X_k^S$. Each arrow represents a journey of less than or equal to $k$ edges in $\mathcal{G}_{[t_1,t_2]}$. Note that each vertex of $S$ is either a vertex of $X_k^S$ or connected by a journey of $k$ edges or less (in either direction) to a vertex of $X_k^S$. This figure illustrates Lemma \ref{['lemma:l6']}.
• Figure 3: A representation of the set S partitioned into three subsets $\{S_1,S_2,S_3\}$ which are connected by journeys of the set $M=\{M_1,M_2\}$ of lengths at most $k$. We also represent the time windows associated with the journeys of $M_1$ and $M_2$ (see the proof \ref{['proof:p4']} for definitions of $S_i$ and $M_i$).This figure illustrates the proof of Lemma \ref{['lemma:l7']}.
• Figure 4: A representation of the concatenated journeys forming the journey described in Lemma \ref{['lemma:l8']} where the vertices of $S$ are represented in green. A blue arrow is a journey traversing at most $n-1$ edges such as defined in Lemma \ref{['lemma:l1']} (journey of type $2$). And a set of green arrows that follow each other is a journey of type $1$ such as defined in Lemma \ref{['lemma:l7']} that traverses at most $2n$ edges and visit between $p$ and $1$ vertices of $S$(see the proof \ref{['proof:p5']} for a definition of $p$). In total these concatenated journeys visit $|S|/2$ vertices of the set $S$.
• Figure 11: A representation of the number of edges traversed in the worst case as a function of the lifetime $L$ for the cycle.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 5
• Definition 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10