Families of tractable problems with respect to vertex-interval-membership width and its generalisations

Computer Science > Discrete Mathematics [Submitted on 21 May 2025 (v1), last revised 15 Jun 2026 (this version, v5)] Title:Families of tractable problems with respect to vertex-interval-membership width and its generalisations View PDF HTML (experimental)Abstract: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. Submission history From: Laura Larios-Jones [view email][v1] Wed, 21 May 2025 16:12:39 UTC (378 KB) [v2] Thu, 25 Sep 2025 13:45:33 UTC (430 KB) [v3] Wed, 22 Oct 2025 10:47:04 UTC (414 KB) [v4] Mon, 27 Apr 2026 16:29:08 UTC (455 KB) [v5] Mon, 15 Jun 2026 20:39:56 UTC (455 KB) Current browse context: cs.DM References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.