Forbidden Patterns in Temporal Graphs Resulting from Encounters in a Corridor

TL;DR

This paper investigates the structure of temporal graphs generated by agents moving in a line or circle, introducing forbidden patterns to characterize realizable graphs and analyzing their properties and recognition algorithms.

Contribution

It introduces the concept of forbidden patterns in temporal graphs derived from mobility models and characterizes which graphs can be realized by agents moving with constant speeds.

Findings

Not all temporal cliques from the 1D model are realizable with distinct constant speeds.

A characterization via circular forbidden patterns is provided for circular mobility models.

A linear-time recognition algorithm is developed for models allowing multiple crossings without finite forbidden patterns.

Abstract

In this paper, we study temporal graphs arising from mobility models, where vertices correspond to agents moving in space and edges appear each time two agents meet. We propose a rather natural one-dimensional model. If each pair of agents meets exactly once, we get a simple temporal clique where the edges are ordered according to meeting times. In order to characterize which temporal cliques can be obtained as such `mobility graphs', we introduce the notion of forbidden patterns in temporal graphs. Furthermore, using a classical result in combinatorics, we count the number of such mobility cliques for a given number of agents, and show that not every temporal clique resulting from the 1D model can be realized with agents moving with different constant speeds. For the analogous circular problem, where agents are moving along a circle, we provide a characterization via circular forbidden patterns. Our characterization in terms of forbidden patterns can be extended to the case where each edge appears at most once. We also study the problem where pairs of agents are allowed to cross each other several times, using an approach from automata theory. We observe that in this case, there is no finite set of forbidden patterns that characterize such temporal graphs and nevertheless give a linear-time algorithm to recognize temporal graphs arising from this model.