On the Treewidth of Dynamic Graphs

TL;DR

This paper investigates the preservation of treewidth in dynamic graphs over time, extending key metatheorems to dynamic settings and demonstrating bounded local treewidth in common dynamic graph classes.

Contribution

It extends Courcelle's and Frick & Grohe's theorems to dynamic graphs with bounded (local) treewidth, enabling new complexity classifications and problem-solving approaches.

Findings

Dynamic graphs of bounded (local) treewidth can be modeled to maintain their treewidth over finite evolution periods.

Certain common dynamic graph classes naturally exhibit bounded local treewidth.

The extended theorems facilitate analysis of dynamic problems using static graph techniques.

Abstract

Dynamic graph theory is a novel, growing area that deals with graphs that change over time and is of great utility in modelling modern wireless, mobile and dynamic environments. As a graph evolves, possibly arbitrarily, it is challenging to identify the graph properties that can be preserved over time and understand their respective computability. In this paper we are concerned with the treewidth of dynamic graphs. We focus on metatheorems, which allow the generation of a series of results based on general properties of classes of structures. In graph theory two major metatheorems on treewidth provide complexity classifications by employing structural graph measures and finite model theory. Courcelle's Theorem gives a general tractability result for problems expressible in monadic second order logic on graphs of bounded treewidth, and Frick & Grohe demonstrate a similar result for first order logic and graphs of bounded local treewidth. We extend these theorems by showing that dynamic graphs of bounded (local) treewidth where the length of time over which the graph evolves and is observed is finite and bounded can be modelled in such a way that the (local) treewidth of the underlying graph is maintained. We show the application of these results to problems in dynamic graph theory and dynamic extensions to static problems. In addition we demonstrate that certain widely used dynamic graph classes naturally have bounded local treewidth.