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Recognizing and Realizing Temporal Reachability Graphs

Thomas Erlebach, Othon Michail, Nils Morawietz

TL;DR

We study Reachability Graph Realizability (RGR) for temporal graphs, asking when a given directed graph $D=(V,A)$ can be realized as the reachability graph of some temporal graph under various path types and labeling restrictions. The authors map a comprehensive complexity landscape: all undirected variants and most directed variants are NP-hard, with two directed variants being trivial; they show polynomial-time solvability when the solid graph is a tree and establish an $\mathrm{FPT}$ algorithm parameterized by the solid graph's feedback edge set number $\mathrm{fes}$. The core techniques include structural analysis of bridge edges, splitting lemmas to decompose instances, LP-based methods for tree-like solids, and an intricate connector framework (robust and nice connectors) enabling a refined FPT approach. The results also prove W[2]-hardness for certain parameters (feedback vertex set and treedepth) and provide ETH-based time lower bounds for the directed case, thereby giving a near-complete picture of the RGR landscape and guiding future kernelization and parameterized approaches for sparse solid graphs.

Abstract

A temporal graph $\mathcal{G}=(G,λ)$ can be represented by an underlying graph $G=(V,E)$ together with a function $λ$ that assigns to each edge $e\in E$ the set of time steps during which $e$ is present. The reachability graph of $\mathcal{G}$ is the directed graph $D=(V,A)$ with $(u,v)\in A$ if only if there is a temporal path from $u$ to $v$. We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph $D=(V,A)$ is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on $λ$ (proper, simple, or happy). Answering an open question posed by Casteigts et al. (Theoretical Computer Science 991 (2024)), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is polynomial-time solvable if the solid graph is a tree and fixed-parameter tractable with respect to the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set or treedepth, since the problem is W[2]-hard with respect to these parameters.

Recognizing and Realizing Temporal Reachability Graphs

TL;DR

We study Reachability Graph Realizability (RGR) for temporal graphs, asking when a given directed graph can be realized as the reachability graph of some temporal graph under various path types and labeling restrictions. The authors map a comprehensive complexity landscape: all undirected variants and most directed variants are NP-hard, with two directed variants being trivial; they show polynomial-time solvability when the solid graph is a tree and establish an algorithm parameterized by the solid graph's feedback edge set number . The core techniques include structural analysis of bridge edges, splitting lemmas to decompose instances, LP-based methods for tree-like solids, and an intricate connector framework (robust and nice connectors) enabling a refined FPT approach. The results also prove W[2]-hardness for certain parameters (feedback vertex set and treedepth) and provide ETH-based time lower bounds for the directed case, thereby giving a near-complete picture of the RGR landscape and guiding future kernelization and parameterized approaches for sparse solid graphs.

Abstract

A temporal graph can be represented by an underlying graph together with a function that assigns to each edge the set of time steps during which is present. The reachability graph of is the directed graph with if only if there is a temporal path from to . We study the Reachability Graph Realizability (RGR) problem that asks whether a given directed graph is the reachability graph of some temporal graph. The question can be asked for undirected or directed temporal graphs, for reachability defined via strict or non-strict temporal paths, and with or without restrictions on (proper, simple, or happy). Answering an open question posed by Casteigts et al. (Theoretical Computer Science 991 (2024)), we show that all variants of the problem are NP-complete, except for two variants that become trivial in the directed case. For undirected temporal graphs, we consider the complexity of the problem with respect to the solid graph, that is, the graph containing all edges that could potentially receive a label in any realization. We show that the RGR problem is polynomial-time solvable if the solid graph is a tree and fixed-parameter tractable with respect to the feedback edge set number of the solid graph. As we show, the latter parameter can presumably not be replaced by smaller parameters like feedback vertex set or treedepth, since the problem is W[2]-hard with respect to these parameters.

Paper Structure

This paper contains 24 sections, 39 theorems, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Organization of the paper.
  4. Preliminaries
  5. The solid graph.
  6. Basic Observations and an Exponential Algorithm
  7. Solid Bridge Edges: Properties and Splitting Operations
  8. Splitting Into Subinstances
  9. Algorithms for Instances with a Tree as Solid Graph
  10. Hardness for Undirected Reachability Graph Realizability
  11. Parameterized hardness
  12. Parameterizing by the Feedback Edge Set Number
  13. Definitions and Notations.
  14. Abstract Description of the Algorithm
  15. Independent structural implication
  16. ...and 9 more sections

Key Result

Lemma 2

Let $D$ be an instance of URGR, and let $e$ be a solid edge of $G$ that is not part of a triangle. In each minimal realization of $D$, $e$ receives at most two labels. Moreover, if $D$ is an instance of any version of URGR besides AnyStrictURGR, then in each minimal realization, $e$ receives at most

Figures (1)

  • Figure 4: An illustration of the main definitions used by the FPT algorithm for parameter $\mathrm{fes}$. The grey area contains the vertices of $V^*$, that is, the $2$-core of $G$.

Theorems & Definitions (50)

  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Definition 6
  • Lemma 7
  • Corollary 8
  • Definition 9
  • Lemma 10
  • Definition 11
  • ...and 40 more