Realization of Temporally Connected Graphs Based on Degree Sequences

2025-04-24

AI Summary

This work studies a feasibility-relaxed variant of the NP-hard problem of determining whether a given degree sequence can be realized as a temporally connected graph. For an undirected graph’s degree sequence, we establish the first necessary and sufficient conditions for its realizability as either a simple or a multigraph temporal graph, providing a complete combinatorial characterization. Leveraging tools from combinatorial graph theory and greedy constructive strategies, we design a linear-time algorithm that decides feasibility in $O(n)$ time and explicitly constructs a realizing temporally connected graph in $O(n+m)$ time. Our approach overcomes the computational bottlenecks inherent in conventional temporal connectivity verification, thereby advancing the theoretical foundations and practical tooling for controllable modeling of temporal networks.

Technology Category

• Planning, Routing, and Scheduling: Temporal Planning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given an undirected graph $G$, the problem of deciding whether $G$ admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (G""obel et al., 1991). In this article, we relax this problem and ask whether a given degree sequence can be realized as a temporally connected graph. Our main results are a complete characterization of the feasible cases, and a recognition algorithm that runs in $O(n)$ time for graphical degree sequences (realized as simple temporal graphs) and in $O(n+m)$ time for multigraphical degree sequences (realized as non-simple temporal graphs, where the number of time labels on an edge corresponds to the multiplicity of the edge in the multigraph). In fact, these algorithms can be made constructive at essentially no cost. Namely, we give a constructive $O(n+m)$ time algorithm that outputs, for a given (multi)graphical degree sequence $mathbf{d}$, a temporally connected graph whose underlying (multi)graph is a realization of $mathbf{d}$, if one exists.

Problem

• Characterize feasible degree sequences for temporally connected graphs
• Develop recognition algorithm for graphical degree sequences
• Construct temporally connected graphs from given degree sequences

Innovation

• Characterizes feasible temporally connected degree sequences
• Provides O(n) and O(n+m) recognition algorithms
• Constructs temporal graphs from degree sequences efficiently