Structural Parameters for Dense Temporal Graphs

TL;DR

This paper introduces temporal analogues of static graph parameters like cliquewidth, modular-width, and neighbourhood diversity, enabling efficient algorithms for certain problems on dense temporal graphs with structured properties.

Contribution

It defines new temporal parameters extending static graph concepts and analyzes their implications for algorithmic tractability in dense temporal graphs.

Findings

Temporal parameters can be small even in dense graphs with many active edges.

Bounded temporal cliquewidth allows efficient solutions for certain problems.

Inclusions among classes based on these parameters are strict under standard complexity assumptions.

Abstract

Temporal graphs provide a useful model for many real-world networks. Unfortunately the majority of algorithmic problems we might consider on such graphs are intractable. There has been recent progress in defining structural parameters which describe tractable cases by simultaneously restricting the underlying structure and the times at which edges appear in the graph. These all rely on the temporal graph being sparse in some sense. We introduce temporal analogues of three increasingly restrictive static graph parameters -- cliquewidth, modular-width and neighbourhood diversity -- which take small values for highly structured temporal graphs, even if a large number of edges are active at each timestep. The computational problems solvable efficiently when the temporal cliquewidth of the input graph is bounded form a subset of those solvable efficiently when the temporal modular-width is bounded, which is in turn a subset of problems efficiently solvable when the temporal neighbourhood diversity is bounded. By considering specific temporal graph problems, we demonstrate that (up to standard complexity theoretic assumptions) these inclusions are strict.