{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T10:57:33Z","timestamp":1758279453233,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For a bar and joint framework $(G,p)$ with point group $\\mathcal{C}_3$ which describes 3-fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946-972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762-773)  that no vertices are fixed by  the automorphism corresponding to the 3-fold rotation (geometrically, no vertices are placed on the center of rotation), are both necessary and sufficient for $(G,p)$ to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups $\\mathcal{C}_2$ and $\\mathcal{C}_s$ which are generated by a half-turn and a reflection in the plane, respectively. In addition, analogously to the results in (Schulze, Discret. Comp. Geom. 44:946-972), we also characterize symmetry generic isostatic graphs for the groups $\\mathcal{C}_2$ and $\\mathcal{C}_s$ in terms of inductive Henneberg-type constructions, as well as Crapo-type 3Tree2 partitions - the full sweep of methods used for the simpler problem without symmetry.<\/jats:p>","DOI":"10.37236\/426","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:59:27Z","timestamp":1578715167000},"source":"Crossref","is-referenced-by-count":11,"title":["Symmetric Laman Theorems for the Groups $\\mathcal{C}_2$ and $\\mathcal{C}_s$"],"prefix":"10.37236","volume":"17","author":[{"given":"Bernd","family":"Schulze","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,11,19]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r154\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r154\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:21:59Z","timestamp":1579303319000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r154"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,11,19]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/426","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,11,19]]},"article-number":"R154"}}