Forbidden Patterns in Temporal Graphs Resulting from Encounters in a Corridor

TL;DR

We address which temporal cliques arise from agents moving in a one-dimensional corridor, modeling edge appearances as encounters. The core method introduces forbidden patterns on triples of nodes to certify 1D-mobility temporal cliques, connects crossing sequences to reduced decompositions of the longest permutation, and yields a linear-time recognition algorithm plus a closed-form counting formula. We further show that these cliques possess linear-size temporal spanners, and we extend the framework to cases with at most one crossing per pair, multi-crossing, and circular mobility, including automaton-based recognizers. The results provide structural insight into mobility-induced temporal graphs and supply efficient tools for recognition, counting, and spanner construction, with avenues for higher-dimensional and more general mobility models.

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.

Figures (9)

• Figure 1: Ordered forbidden patterns in an ordered 1D-mobility temporal clique. Each pattern is ordered from left to right and has associated ordering $a,b,c$.
• Figure 2: Relative order of labels of edges between the sets $V_L$, $V_R$ and the two vertices $v_1$ and $v_n$, showing how temporal paths between $V_L$ and $V_R$ can be formed.
• Figure 3: Any 1D-mobility clique with the above prefix cannot be realized with agents moving at constant speed.
• Figure 4: Ordered forbidden patterns with forbidden edges in the 1D-mobility model when each pair of agents cross at most once. The ordering associated to each pattern is $a,b,c$.
• Figure 5: Automaton of possible edge sequences on the nodes $a$, $b$ and $c$.

Theorems & Definitions (38)

• Theorem 1
• Proposition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 2
• proof
• Proposition 2
• Lemma 3