Temporal graphs are graphs whose edges are labelled with times at which they are active. Their time-sensitivity provides a useful model of real networks, but renders many problems studied on temporal graphs more computationally complex than their static counterparts. To contend with this, there has been recent work devising parameters for which temporal problems become tractable. One such parameter is vertex-interval-membership (VIM) width. Broadly, this gives a bound on the number of vertices we need to keep track of at any given time to solve many problems. Our contributions are two-fold. Firstly, we introduce a new parameter, tree-interval-membership (TIM) width, that generalises both VIM width and several existing generalisations. Secondly, we provide meta-algorithms for both VIM and TIM width which can be used to prove fixed-parameter-tractability for large families of problems, bypassing the need to give involved dynamic programming arguments for every problem. In doing this, we provide a characterisation of problems in FPT with respect to both parameters. We apply these algorithms to temporal versions of Hamiltonian path, dominating set, matching, and edge deletion to limit maximum reachability.

翻译：时态图是一种边被标记为活动时间的图。其时间敏感性为真实网络提供了有用的模型，但也使得许多在时态图上研究的问题比其静态对应问题在计算上更为复杂。为应对这一挑战，近期研究致力于设计使时态问题变得可处理的参数。其中一个参数是顶点区间成员度宽度。简而言之，该参数界定了在任意给定时间需要跟踪的顶点数量以解决许多问题。我们的贡献有两个方面。首先，我们引入了一个新参数——树区间成员度宽度，该参数同时推广了顶点区间成员度宽度和若干现有推广形式。其次，我们为顶点区间成员度宽度和树区间成员度宽度提供了元算法，可用于证明大量问题族的固定参数可处理性，从而避免了为每个问题构建复杂的动态规划论证。在此过程中，我们刻画了相对于这两个参数属于固定参数可处理类的问题特征。我们将这些算法应用于时态版本的哈密顿路径、支配集、匹配以及限制最大可达性的边删除问题。