Forbidden patterns in temporal graphs resulting from encounters in a corridor

Abstract

In this paper, we study temporal graphs arising from 1 dimensional mobility models, where vertices correspond to agents moving on a line and edges appear each time two agents meet. If each pair of agents meets exactly once, we get a simple temporal clique. In order to characterize such temporal cliques, we introduce the notion of forbidden patterns in temporal graphs. We extend the forbidden pattern characterization to simple mobility graphs (where each edge appears at most once) and to the analogous circular problem, where agents are moving along a circle. 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, nevertheless provide a linear time algorithm to recognize them.