We study the parameterized complexity of maximum temporal connected components (tccs) in temporal graphs, i.e., graphs that deterministically change over time. In a tcc, any pair of vertices must be able to reach each other via a time-respecting path. We consider both problems of maximum open tccs (openTCC), which allow temporal paths through vertices outside the component, and closed tccs (closedTCC) which require at least one temporal path entirely within the component for every pair. We focus on the structural parameter of treewidth, tw, and the recently introduced temporal parameter of temporal path number, tpn, which is the minimum number of paths needed to fully describe a temporal graph. We prove that these parameters on their own are not sufficient for fixed parameter tractability: both openTCC and closedTCC are NP-hard even when tw=9, and closedTCC is NP-hard when tpn=6. In contrast, we prove that openTCC is in XP when parameterized by tpn. On the positive side, we show that both problem become fixed parameter tractable under various combinations of structural and temporal parameters that include, tw plus tpn, tw plus the lifetime of the graph, and tw plus the maximum temporal degree.

翻译：本文研究时态图中最大时态连通分量（tcc）的参数化复杂度。时态图指随时间确定性变化的图结构。在时态连通分量中，任意顶点对必须能通过时间一致性路径相互可达。我们同时考察允许穿越分量外顶点的最大开放时态连通分量（openTCC）问题，以及要求任意顶点对至少存在完全位于分量内的时间一致性路径的封闭时态连通分量（closedTCC）问题。研究聚焦于结构参数树宽（tw）与新近提出的时态参数时态路径数（tpn）——该参数指完整描述时态图所需的最小路径数量。我们证明这些参数单独作用时不足以实现固定参数可解性：当tw=9时，openTCC与closedTCC均为NP难问题；当tpn=6时，closedTCC亦为NP难问题。与之相对，我们证明openTCC在tpn参数化下属于XP类。从积极角度看，我们证实在多种结构与时态参数的组合下，包括（tw+tpn）、（tw+图生命周期）以及（tw+最大时态度数），两类问题均可实现固定参数可解。