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Structural Parameters for Dense Temporal Graphs

Jessica Enright, Samuel D. Hand, Laura Larios-Jones, Kitty Meeks

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.

Structural Parameters for Dense Temporal Graphs

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.
Paper Structure (9 sections, 29 theorems, 2 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 29 theorems, 2 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Tractability with respect to Temporal Cliquewidth
  4. Tractability with respect to Temporal Modular-width
  5. Star Exploration
  6. Tractability with respect to Temporal Neighbourhood Diversity
  7. Temporal Graph Burning
  8. Minimum Reachability Edge Deletion
  9. Conclusion and Open Questions

Key Result

Proposition 3

If a temporal graph $\mathcal{G}$ has temporal cliquewidth $k$, then the $\Delta$-association graph $G$ of $\mathcal{G}$ has cliquewidth at most $k$.

Figures (1)

  • Figure 1: A diagram of our parameters and the problems we show to be tractable with respect to each. A problem is in a rectangle if it is tractable with respect to the parameter it is labelled with. Assuming P$\neq$NP, each problem is in the rectangle for the most general of the three parameters for which it is tractable.

Theorems & Definitions (45)

  • Definition 1: Cliquewidth
  • Definition 2: Temporal Cliquewidth
  • Proposition 3
  • Theorem 4
  • Definition 5: Modular-width, Section 2.5 gajarsky_parameterized_2013
  • Definition 6: Temporal Modular-width
  • Theorem 7
  • Definition 8: Splitter DBLP:journals/csr/HabibP10
  • Definition 9: Static Module DBLP:journals/csr/HabibP10
  • Definition 10: Temporal Module
  • ...and 35 more