{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T11:45:42Z","timestamp":1753357542996,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,8,19]],"date-time":"2023-08-19T00:00:00Z","timestamp":1692403200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,8,19]],"date-time":"2023-08-19T00:00:00Z","timestamp":1692403200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National University Ireland, Galway"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer <jats:italic>b<\/jats:italic> there is such an inductive construction of triangulations with <jats:italic>b<\/jats:italic> braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with <jats:italic>b<\/jats:italic> braces that is linear in\u00a0<jats:italic>b<\/jats:italic>. In the case that <jats:inline-formula><jats:alternatives><jats:tex-math>$$b=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>b<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and a class of mixed norms on\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-023-00546-5","type":"journal-article","created":{"date-parts":[[2023,8,19]],"date-time":"2023-08-19T19:02:34Z","timestamp":1692471754000},"page":"1238-1275","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Braced Triangulations and Rigidity"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4731-9302","authenticated-orcid":false,"given":"James","family":"Cruickshank","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Eleftherios","family":"Kastis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Derek","family":"Kitson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Bernd","family":"Schulze","sequence":"additional","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,8,19]]},"reference":[{"issue":"1","key":"546_CR1","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/BF02764905","volume":"67","author":"DW Barnette","year":"1989","unstructured":"Barnette, D.W., Edelson, A.L.: All $$2$$-manifolds have finitely many minimal triangulations. 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