AI 中文总结
该研究将图的角色着色扩展到时序图，证明时序角色着色问题是NP完全的，并通过参数化分析给出了该问题的固定参数可处理性结果。
AI 中文摘要
图$G$的角色着色是对$G$的顶点分配颜色，使得同色顶点的邻域拥有完全相同的颜色集合，该模型用于刻画接触网络中顶点具有角色的理念，与接触对象的角色集一致。我们将角色着色问题扩展到时序图，通过自动机定义时序角色，自动机的状态和转移捕捉顶点当前颜色以及当前与过去邻接的信息。我们通过归约到静态问题证明，时序角色着色问题是NP完全的。为应对该难解性，我们探索了多种参数化方式：给出了结合自动机状态数与时序图的顶点区间成员宽度或树区间成员宽度的固定参数可处理性结果；进一步证明，结合基础图的树宽、时序图的生命周期和颜色数作为参数时，该问题属于固定参数可处理（FPT）类。
英文摘要
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.
