Exploration of Always $S$-Connected Temporal Graphs

TL;DR

This paper introduces the concept of always $S$-connected temporal graphs to analyze exploration problems, providing new algorithms and bounds for exploring such graphs with multiple or single agents, especially in graphs with bounded treewidth or specific clique structures.

Contribution

It generalizes connectedness in temporal graphs to $S$-connectedness, and develops exploration algorithms with improved bounds for graphs with bounded treewidth and specific clique properties.

Findings

Exploration of $S$-connected temporal graphs with $m=|S|$ agents in $O(n^{1.5} m^3 riangle^{1.5} ext{log}^{1.5}(n))$ snapshots.

Single-agent exploration bounds for graphs with bounded treewidth, improving previous results.

Efficient exploration of interval graphs with few large cliques in $O(n^{4/3} ext{log}^{2.5}(n))$ snapshots.

Abstract

\emph{Temporal graphs} are a generalisation of (static) graphs, defined by a sequence of \emph{snapshots}, each a static graph defined over a common set of vertices. \emph{Exploration} problems are one of the most fundamental and most heavily studied problems on temporal graphs, asking if a set of $m$ agents can visit every vertex in the graph, with each agent only allowed to traverse a single edge per snapshot. In this paper, we introduce and study \emph{always $S$-connected} temporal graphs, a generalisation of always connected temporal graphs where, rather than forming a single connected component in each snapshot, we have at most $\vert S \vert$ components, each defined by the connection to a single vertex in the set $S$. We use this formulation as a tool for exploring graphs admitting an \emph{$(r,b)$-division}, a partitioning of the vertex set into disconnected components, each of which is $S$-connected, where $\vert S \vert \leq b$. We show that an always $S$-connected temporal graph with $m = \vert S \vert$ and an average degree of $\Delta$ can be explored by $m$ agents in $O(n^{1.5} m^3 \Delta^{1.5}\log^{1.5}(n))$ snapshots. Using this as a subroutine, we show that any always-connected temporal graph with treewidth at most $k$ can be explored by a single agent in $O\left(n^{4/3} k^{5.5}\log^{2.5}(n)\right)$ snapshots, improving on the current state-of-the-art for small values of $k$. Further, we show that interval graph with only a small number of large cliques can be explored by a single agent in $O\left(n^{4/3} \log^{2.5}(n)\right)$ snapshots.