Exploring Word-Representable Temporal Graphs
Duncan Adamson
TL;DR
The paper investigates the exploration problem on word-representable temporal graphs, where edges are determined by alternating symbol occurrences in a representing word. By leveraging spanning-tree based traversal and analyzing waiting times via graph parameters, the authors establish a $2\delta n$ upper bound for exploration on always-connected graphs and a $2 d n$ upper bound for general word-representable graphs with a long enough word relative to diameter $d$. They also prove asymptotic lower bounds of $\Omega(d n)$, demonstrating near-tightness of the upper bounds. The work combines structural lemmas on word representations with temporal traversal techniques, advancing understanding of exploration in dynamic, word-representable networks.
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 $2δn$ timesteps, where $δ$ 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 $Ω(d n)$ timesteps to explore, for any $d \in [1, n]$.