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Exploration of Always $S$-Connected Temporal Graphs

Duncan Adamson, Paul G Spirakis

TL;DR

This paper introduces and studies always connected temporal graphs, a generalisation of always connected temporal graphs where, rather than forming a single connected component in each snapshot, the authors have at most at most $\vert S \vert$ components, each defined by the connection to a single vertex in the set $S$.

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 $Δ$ can be explored by $m$ agents in $O(n^{1.5} m^3 Δ^{1.5}\log^{1.5}(n))$ 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.

Exploration of Always $S$-Connected Temporal Graphs

TL;DR

This paper introduces and studies always connected temporal graphs, a generalisation of always connected temporal graphs where, rather than forming a single connected component in each snapshot, the authors have at most at most components, each defined by the connection to a single vertex in the set .

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 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 -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 components, each defined by the connection to a single vertex in the set . We use this formulation as a tool for exploring graphs admitting an \emph{-division}, a partitioning of the vertex set into disconnected components, each of which is -connected, where . We show that an always -connected temporal graph with and an average degree of can be explored by agents in snapshots. Using this as a subroutine, we show that any always-connected temporal graph with treewidth at most can be explored by a single agent in snapshots, improving on the current state-of-the-art for small values of . Further, we show that interval graph with only a small number of large cliques can be explored by a single agent in snapshots.
Paper Structure (8 sections, 19 theorems)

This paper contains 8 sections, 19 theorems.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Tools
  5. Lower bounds on Exploring Always $S$-Connected Graphs
  6. Exploring S-Always Connected Graphs
  7. Applications
  8. Conclusion

Key Result

Theorem 1

Let $\mathcal{G} = G_1, G_2, \dots, G_T$ be an always $S$-connected temporal graph with lifetime $T = O\left( n m^3 \Delta^{1.5} \sqrt{\vert X \vert \log\left(m\vert X \vert\right)}\log(\vert X \vert)\right)$ where $m = \vert S \vert$ and $\Delta$ is the average degree of the graph. Then any subset

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Definition 3: Always $S$-Connected Temporal Graphs
  • Lemma 4: Reachability, Lemma 2.1 in erlebach2021temporal
  • Lemma 5: Multi to Single Agent Exploration, Lemma 2.2 in erlebach2021temporal
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 11 more