The Parameterised Complexity of Temporal Motif Counting, and a Lovász-Style Isomorphism Theorem

2026-07-09

AI Summary

This work investigates the homomorphism counting problem for temporal patterns with partially ordered edges in large temporal graphs, aiming to characterize the structural expressiveness of temporal graphs. By establishing an equivalence between temporal graph isomorphism and homomorphism counts of temporal patterns, the study formulates a temporal analogue of Lovász’s isomorphism theorem. It introduces a novel width parameter, termed *toadwidth*, to analyze fixed-parameter tractability. Combining techniques from parameterized complexity, extensions of clique-width, and combinatorial graph theory, the paper proves that two temporal graphs are isomorphic if and only if they admit identical homomorphism counts for all temporal patterns; that homomorphism counting is fixed-parameter tractable when parameterized by bounded toadwidth; and provides a sharp dichotomy criterion for the parameterized complexity of homomorphism counting in the case of totally ordered temporal patterns.

Abstract

We study the structural expressivity and the parameterised complexity of counting homomorphisms from small temporal patterns to large temporal graphs. Here, a temporal pattern $P$ consists of a graph together with a partial order on its edges, and a homomorphism from $P$ to a temporal graph must not only preserve edges, but also satisfy the temporal constraints imposed by the partial order of the edge set of the pattern. The main results of this work are three-fold: First, we prove a temporal Lovász-style theorem, stating that two temporal graphs are isomorphic (under a natural definition of temporal isomorphisms) if and only if they have the same number of homomorphisms from all temporal patterns. Second, we introduce a cliquewidth-based measure on temporal patterns, called the temporally order-augmented dual width, the "toadwidth" for short, and show that counting temporal homomorphisms is fixed-parameter tractable for temporal patterns of bounded toadwidth. Third, we provide a parameterised complexity dichotomy with an explicit tractability criterion for counting homomorphisms from totally ordered temporal patterns, classified along their underlying graph structure.

Problem

• temporal motif counting
• parameterised complexity
• temporal graphs
• homomorphism counting
• temporal patterns

Innovation

• temporal motif counting
• Lovász-style theorem
• toadwidth
• parameterized complexity
• temporal homomorphism