FO and MSO Model Checking on Temporal Graphs

TL;DR

This paper extends logical meta-theorems to temporal graphs by introducing new parameters and encodings, enabling fixed-parameter tractability results for a broad class of problems.

Contribution

It develops a unifying logical framework for temporal graphs, introducing VIM and TIM width parameters, and extends MSO and FO meta-theorems to these parameters.

Findings

Extended MSO meta-theorems for bounded lifetime and degree.

First FO meta-theorems for VIM and TIM widths.

A modular FO and MSO formula lexicon for temporal graph problems.

Abstract

Algorithmic meta-theorems provide an important tool for showing tractability of graph problems on graph classes defined by structural restrictions. While such results are well established for static graphs, corresponding frameworks for temporal graphs are comparatively limited. In this work, we revisit past applications of logical meta-theorems to temporal graphs and develop an extended unifying logical framework. Our first contribution is the introduction of logical encodings for the parameters vertex-interval-membership (VIM) width and tree-interval-membership (TIM) width, parameters which capture the signature of vertex and component activity over time. Building on this, we extend existing monadic second-order (MSO) meta-theorems for bounded lifetime and temporal degree to the parameters VIM and TIM width, and establish novel first-order (FO) meta-theorems for all four parameters. Finally, we signpost a modular lexicon of reusable FO and MSO formulas for a broad range of temporal graph problems, and give an example. This lexicon allows new problems to be expressed compositionally and directly yields fixed-parameter tractability results across the four parameters we consider.