Vertex cuts and median decompositions

Mathematics > Combinatorics [Submitted on 17 Jun 2026] Title:Vertex cuts and median decompositions View PDF HTML (experimental)Abstract:Median decompositions were introduced by Stavropoulos in 2015 as a generalisation of tree decompositions. In this paper, we further develop and exposit this theory as a tool in structural graph theory to study systems of vertex separations. Generalising the well-known fact that nested systems of vertex separations produce tree decompositions of a graph over the structure tree, we describe how a (not necessarily nested) system of separations produces a median decomposition. The median graph in this decomposition is the `dual median graph' constructed by Sageev. If the system of cuts is nested then this median decomposition recovers precisely the aforementioned tree decomposition. We prove a theorem asserting that this decomposition is `uniquely minimal', and describe how Sageev--Roller duality manifests in median decompositions. As an application of our structural approach, we extend a theorem of Stavropoulos from finite graphs to all graphs, which states that the median-width a graph is equal to its clique number. We also describe the link between (canonical) median decompositions and (equivariant) coarse embeddings/quasi-isometries into median graphs. A corollary of these results is a characterisation of when a finitely generated group acts metrically-properly/geometrically on a median graph, in terms of canonical median decompositions of its Cayley graphs. 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.