{"date_updated":"2023-09-15T12:19:32Z","month":"01","article_processing_charge":"No","day":"01","type":"conference","abstract":[{"text":"We present an efficient algorithm for a problem in the interface between clustering and graph embeddings. An embedding ' : G ! M of a graph G into a 2manifold M maps the vertices in V (G) to distinct points and the edges in E(G) to interior-disjoint Jordan arcs between the corresponding vertices. In applications in clustering, cartography, and visualization, nearby vertices and edges are often bundled to a common node or arc, due to data compression or low resolution. This raises the computational problem of deciding whether a given map ' : G ! M comes from an embedding. A map ' : G ! M is a weak embedding if it can be perturbed into an embedding ψ: G ! M with k' \"k < \" for every &quot; &gt; 0. A polynomial-time algorithm for recognizing weak embeddings was recently found by Fulek and Kyncl [14], which reduces to solving a system of linear equations over Z2. It runs in O(n2!) O(n4:75) time, where 2:373 is the matrix multiplication exponent and n is the number of vertices and edges of G. We improve the running time to O(n log n). Our algorithm is also conceptually simpler than [14]: We perform a sequence of local operations that gradually &quot;untangles&quot; the image '(G) into an embedding (G), or reports that ' is not a weak embedding. It generalizes a recent technique developed for the case that G is a cycle and the embedding is a simple polygon [1], and combines local constraints on the orientation of subgraphs directly, thereby eliminating the need for solving large systems of linear equations.","lang":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/1709.09209","open_access":"1"}],"department":[{"_id":"UlWa"}],"publication_status":"published","date_published":"2018-01-01T00:00:00Z","scopus_import":"1","publisher":"ACM","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","date_created":"2018-12-11T11:45:45Z","oa":1,"oa_version":"Preprint","related_material":{"record":[{"relation":"later_version","id":"6982","status":"public"}]},"status":"public","quality_controlled":"1","title":"Recognizing weak embeddings of graphs","external_id":{"isi":["000483921200021"],"arxiv":["1709.09209"]},"page":"274 - 292","doi":"10.1137/1.9781611975031.20","author":[{"full_name":"Akitaya, Hugo","last_name":"Akitaya","first_name":"Hugo"},{"orcid":"0000-0001-8485-1774","full_name":"Fulek, Radoslav","id":"39F3FFE4-F248-11E8-B48F-1D18A9856A87","first_name":"Radoslav","last_name":"Fulek"},{"first_name":"Csaba","last_name":"Tóth","full_name":"Tóth, Csaba"}],"language":[{"iso":"eng"}],"_id":"309","citation":{"mla":"Akitaya, Hugo, et al. <i>Recognizing Weak Embeddings of Graphs</i>. ACM, 2018, pp. 274–92, doi:<a href=\"https://doi.org/10.1137/1.9781611975031.20\">10.1137/1.9781611975031.20</a>.","ieee":"H. Akitaya, R. Fulek, and C. Tóth, “Recognizing weak embeddings of graphs,” presented at the SODA: Symposium on Discrete Algorithms, New Orleans, LA, USA, 2018, pp. 274–292.","ista":"Akitaya H, Fulek R, Tóth C. 2018. Recognizing weak embeddings of graphs. SODA: Symposium on Discrete Algorithms, 274–292.","apa":"Akitaya, H., Fulek, R., &#38; Tóth, C. (2018). Recognizing weak embeddings of graphs (pp. 274–292). Presented at the SODA: Symposium on Discrete Algorithms, New Orleans, LA, USA: ACM. <a href=\"https://doi.org/10.1137/1.9781611975031.20\">https://doi.org/10.1137/1.9781611975031.20</a>","chicago":"Akitaya, Hugo, Radoslav Fulek, and Csaba Tóth. “Recognizing Weak Embeddings of Graphs,” 274–92. ACM, 2018. <a href=\"https://doi.org/10.1137/1.9781611975031.20\">https://doi.org/10.1137/1.9781611975031.20</a>.","ama":"Akitaya H, Fulek R, Tóth C. Recognizing weak embeddings of graphs. In: ACM; 2018:274-292. doi:<a href=\"https://doi.org/10.1137/1.9781611975031.20\">10.1137/1.9781611975031.20</a>","short":"H. Akitaya, R. Fulek, C. Tóth, in:, ACM, 2018, pp. 274–292."},"conference":{"end_date":"2018-01-10","location":"New Orleans, LA, USA","start_date":"2018-01-07","name":"SODA: Symposium on Discrete Algorithms"},"year":"2018","publist_id":"7556","project":[{"call_identifier":"FWF","name":"Eliminating intersections in drawings of graphs","_id":"261FA626-B435-11E9-9278-68D0E5697425","grant_number":"M02281"}],"acknowledgement":"∗Research supported in part by the NSF awards CCF-1422311 and CCF-1423615, and the Science Without Borders program. The second author gratefully acknowledges support from Austrian Science Fund (FWF): M2281-N35.","isi":1}