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For example, we ask whether every geodesic metric space (or graph) without a fat <jats:italic>H<\/jats:italic> minor is quasi-isometric to a graph with no <jats:italic>H<\/jats:italic> minor, for an arbitrary finite graph <jats:italic>H<\/jats:italic>. We answer this affirmatively for a few small <jats:italic>H<\/jats:italic>. We also present a metric analogue of Menger\u2019s theorem and K\u00f6nig\u2019s ray theorem. We conjecture metric analogues of the Erd\u0151s\u2013P\u00f3sa Theorem and Halin\u2019s grid theorem.<\/jats:p>","DOI":"10.1007\/s00493-025-00150-6","type":"journal-article","created":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T09:26:48Z","timestamp":1749806808000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Graph Minors and Metric Spaces"],"prefix":"10.1007","volume":"45","author":[{"given":"Agelos","family":"Georgakopoulos","sequence":"first","affiliation":[]},{"given":"Panos","family":"Papasoglu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,13]]},"reference":[{"issue":"2","key":"150_CR1","doi-asserted-by":"publisher","first-page":"1438","DOI":"10.1137\/23M1573082","volume":"38","author":"S Albrechtsen","year":"2024","unstructured":"Albrechtsen, S., Huynh, T., Jacobs, R.W., Knappe, P., Wollan, P.: A Menger-type theorem for two induced paths. 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