Temporal Role Colouring

2026-07-31

AI Summary

This study addresses the problem of role coloring in temporal graphs, where vertices are assigned colors such that vertices sharing the same color exhibit identical role sets within their current and historical neighborhoods. As the first temporal extension of role coloring from static graphs, this work introduces a temporal role coloring model and employs automata to characterize the evolution of vertex roles over time. Theoretical analysis establishes that the problem is NP-complete. Furthermore, leveraging structural parameters such as treewidth and vertex-interval membership width, the authors develop fixed-parameter tractable algorithms that yield efficient solutions under various parameter combinations.

Abstract

A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. This model is used to capture the idea of vertices having roles in a contact network, consistent with the set of roles of their contacts. We define an extension of the role colouring problem to temporal graphs. Temporal roles are defined via an automaton with states and transitions capturing both the current colour of a vertex and information about its current and past adjacencies. We show, by a reduction from the static problem, that the temporal role colouring problem is NP-complete. To contend with this intractability, we explore several parameterisations. We give fixed-parameter tractability results with respect to the number of states of the automaton combined with either the vertex-interval-membership width or the tree-interval-membership width of the temporal graph. We further show the problem is in FPT parameterised by the treewidth of the underlying graph, the lifetime of the temporal graph and the number of colours combined.

Problem

• temporal role colouring
• temporal graphs
• role colouring
• NP-completeness
• parameterised complexity

Innovation

• temporal role colouring
• parameterized complexity
• fixed-parameter tractability
• temporal graphs
• automaton-based modeling