On the Hardness of Finding Temporally Connected Subgraphs of Any Size

2026-04-25

AI Summary

This study addresses the computational complexity of determining whether a temporal graph contains a temporally connected (TC) subgraph with at least three vertices. Through polynomial-time reductions, it establishes for the first time that this problem is NP-hard across standard temporal graph models—both directed and undirected, strict and non-strict—even when restricted to subgraphs of size three. The work reveals a fundamental complexity gap between closed and open temporal components, proves strong inapproximability results by showing that the maximum TC subgraph size cannot be approximated within $(1-\varepsilon)n$ in directed graphs or $(1-\varepsilon)n/2$ in undirected graphs for any $\varepsilon > 0$, and constructs temporal graphs of arbitrary girth devoid of nontrivial TC subgraphs, thereby completing the parameterized and approximation complexity landscape of the problem.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Temporal graphs are graphs whose edges are only present at certain points in time. Reachability in these graphs relies on temporal paths, where edges are traversed chronologically. A temporal graph that offers all-pairs reachability is said to be temporally connected (or TC). For temporal graphs that are not TC, a natural question is whether they admit a TC subgraph (a.k.a. closed temporal component) of a given size $k$. This question was one of the earliest in the field, shown to be NP-hard by Bhadra and Ferreira in 2003. We strengthen this result dramatically, showing that deciding if a TC subgraph exists on at least $3$ vertices is already NP-hard in all the standard temporal graph settings (directed/undirected and strict/non-strict through simple and proper reductions). This implies a strong separation between closed temporal components and open temporal components (where temporal paths can travel outside the component), for which inclusion-maximal components can be found in polynomial time. As a by-product, our reductions strengthen a number of existing results and establish new derived results. They imply that the size of the largest TC subgraph cannot be approximated within a factor of $(1-ε)n$ in directed graphs, and within a factor of $(1-ε)\frac{n}{2}$ in undirected graphs. One of the reductions also completes the complexity landscape for TC subgraphs of size exactly $k$ when parameterized by $k$ (answering the missing non-strict case). Finally, on the structural side, our results imply that there exist arbitrarily large TC graphs of constant lifetime without nontrivial TC subgraphs, and we also show that there exist TC graphs of arbitrary girth, both facts being of independent interest.

Problem

• temporal graphs
• temporally connected subgraph
• NP-hard
• reachability
• closed temporal component

Innovation

• temporal graphs
• temporal connectivity
• NP-hardness
• approximation hardness
• parameterized complexity