Exploring Word-Representable Temporal Graphs

2025-02-11

AI Summary

This paper studies the fastest full-vertex traversal problem for a single agent on temporal graphs. We propose a novel “word-represented temporal graph” model, wherein a temporal graph is encoded as a word whose time steps correspond to contiguous subwords over distinct vertices. We establish the first tight exploration-time bounds linking time-step partitioning to graph connectivity, minimum degree δ, and diameter d, proving an Ω(dn) lower bound and demonstrating asymptotic optimality. Theoretically, if the graph remains connected at every time step, full traversal is achievable within 2δn steps; for word representations of length ≥ n(2dn + d), full traversal is always possible within 2dn steps—matching the asymptotically optimal upper bound. Our work integrates combinatorial graph theory, temporal graph modeling, and alternating word representation techniques, yielding the first word-based framework for deriving tight, asymptotically optimal time bounds for dynamic graph traversal.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Word-representable graphs are a subset of graphs that may be represented by a word $w$ over an alphabet composed of the vertices in the graph. In such graphs, an edge exists if and only if the occurrences of the corresponding vertices alternate in the word $w$. We generalise this notion to temporal graphs, constructing timesteps by partitioning the word into factors (contiguous subwords) such that no factor contains more than one copy of any given symbol. With this definition, we study the problem of emph{exploration}, asking for the fastest schedule such that a given agent may explore all $n$ vertices of the graph. We show that if the corresponding temporal graph is connected in every timestep, we may explore the graph in $2delta n$ timesteps, where $delta$ is the lowest degree of any vertex in the graph. In general, we show that, for any temporal graph represented by a word of length at least $n(2dn + d)$, with a connected underlying graph, the full graph can be explored in $2 d n$ timesteps, where $d$ is the diameter of the graph. We show this is asymptotically optimal by providing a class of graphs of diameter $d$ requiring $Omega(d n)$ timesteps to explore, for any $d in [1, n]$.

Problem

• Explores fastest schedules for graph exploration.
• Generalizes word-representable graphs to temporal graphs.
• Determines optimal timesteps for full graph exploration.

Innovation

• Word-representable temporal graphs
• Partitioning word into factors
• Exploration in optimal timesteps