{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:54:48Z","timestamp":1760144088443,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T00:00:00Z","timestamp":1710892800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministrstvo za visoko \u0161olstvo, znanost in tehnologijo Slovenije","award":["P1-0285"],"award-info":[{"award-number":["P1-0285"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A graphoid is a mixed multigraph with multiple directed and\/or undirected edges, loops, and semiedges. A covering projection of graphoids is an onto mapping between two graphoids such that at each vertex, the mapping restricts to a local bijection on incoming edges and outgoing edges. Naturally, as it appears, this definition displays unusual behaviour since the projection of the corresponding underlying graphs is not necessarily a graph covering. Yet, it is still possible to grasp such coverings algebraically in terms of the action of the fundamental monoid and combinatorially in terms of voltage assignments on arcs. In the present paper, the existence theorem is formulated and proved in terms of the action of the fundamental monoid. A more conventional formulation in terms of the weak fundamental group is possible because the action of the fundamental monoid is permutational. The standard formulation in terms of the fundamental group holds for a restricted class of coverings, called homogeneous. Further, the existence of the universal covering and the problems related to decomposing regular coverings via regular coverings are studied in detail. It is shown that with mild adjustments in the formulation, all the analogous theorems that hold in the context of graphs are still valid in this wider setting.<\/jats:p>","DOI":"10.3390\/sym16030375","type":"journal-article","created":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T13:09:36Z","timestamp":1710940176000},"page":"375","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Coverings of Graphoids: Existence Theorem and Decomposition Theorems"],"prefix":"10.3390","volume":"16","author":[{"given":"Aleksander","family":"Malni\u010d","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Education, University of Ljubljana, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia"},{"name":"Department of Mathematics, IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boris","family":"Zgrabli\u0107","sequence":"additional","affiliation":[{"name":"Department of Mathematics, FAMNIT, Univerza na Primorskem, Glagolja\u0161ka 8, 6000 Koper, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,20]]},"reference":[{"key":"ref_1","unstructured":"Malni\u010d, A., and Zgrabli\u0107, B. (2023). Covers of general digraphs. Ars Math. Contemp., submitted for publication."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Biggs, N.L. (1974). Algebraic Graph Theory, Cambridge University Press.","DOI":"10.1017\/CBO9780511608704"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/0095-8956(74)90070-7","article-title":"Automorphisms of graphs and coverings","volume":"16","year":"1974","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0012-365X(74)90006-5","article-title":"Voltage graphs","volume":"9","author":"Gross","year":"1974","journal-title":"Discret. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/0012-365X(79)90181-X","article-title":"Observations on the construction of covers using permutation voltage assignments","volume":"28","author":"Ezell","year":"1979","journal-title":"Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/jlms\/s2-30.1.1","article-title":"Homological coverings of graphs","volume":"30","author":"Biggs","year":"1984","journal-title":"J. Lond. Math. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/0012-365X(86)90099-3","article-title":"A contribution to the theory of voltage graphs","volume":"61","year":"1986","journal-title":"Discret. Math."},{"key":"ref_8","unstructured":"Gross, J.L., and Tucker, T.W. (1987). Topological Graph Theory, Wiley\u2013Interscience."},{"key":"ref_9","first-page":"113","article-title":"Constructing and forbidding automorphisms in lifted maps","volume":"47","author":"Archdeacon","year":"1997","journal-title":"Math. Slovaca"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2977","DOI":"10.1090\/mcom\/3637","article-title":"The simultaneous conjugacy problem in the symmetric group","volume":"90","author":"Brodnik","year":"2021","journal-title":"Math. Comp."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/j.jalgebra.2013.02.035","article-title":"Arc-transitive abelian regular covers of cubic graphs","volume":"387","author":"Conder","year":"2013","journal-title":"J. Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1006\/jctb.1998.1847","article-title":"On 2-arc-transitive covers of complete graphs","volume":"74","author":"Du","year":"1998","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s10801-014-0542-5","article-title":"Elementary abelian regular coverings of Platonic maps. Case I: Ordinary representations","volume":"41","author":"Jones","year":"2015","journal-title":"J. Algebr. Comb."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/S0195-6698(03)00055-6","article-title":"s-Regular cyclic coverings of the three-dimensional hypercube Q3","volume":"24","author":"Feng","year":"2003","journal-title":"Eur. J. Comb."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1142\/S0252959904000056","article-title":"s-Regular dihedral coverings of the complete graph of order 4","volume":"25","author":"Feng","year":"2004","journal-title":"Chin. Ann. Math. B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3647","DOI":"10.1090\/S0002-9947-07-04040-8","article-title":"Semi-edges, reflections and Coxeter groups","volume":"359","author":"Gramlich","year":"2007","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1137\/S0895480193248186","article-title":"Enumeration of concrete regular covering projections","volume":"8","author":"Hofmeister","year":"1995","journal-title":"SIAM J. Discret. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.jctb.2018.01.007","article-title":"Tetravalent vertex-and edge-transitive graphs over doubled cycles","volume":"131","author":"Kuzman","year":"2018","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_19","unstructured":"Li, C.H., and Zhu, Y.Z. (2022). Covers and pseudocovers of symmetric graphs. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1006\/eujc.2000.0390","article-title":"Lifting graph automorphisms by voltage assignments","volume":"21","author":"Nedela","year":"2000","journal-title":"Eur. J. Comb."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1023\/B:JACO.0000047294.42633.25","article-title":"Elementary abelian covers of graphs","volume":"20","year":"2004","journal-title":"J. Algebr. Comb."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1016\/j.jcta.2016.04.003","article-title":"Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8","volume":"145","year":"2017","journal-title":"J. Comb. Theory Ser. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"393","DOI":"10.26493\/1855-3974.373.4cd","article-title":"Sectional split extensions arising from lifts of groups","volume":"6","year":"2013","journal-title":"Ars Math. Contemp."},{"key":"ref_24","first-page":"147","article-title":"Testing whether the lifted group splits","volume":"11","year":"2016","journal-title":"Ars Math. Contemp."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.jsc.2017.11.017","article-title":"Computing stable epimorphisms onto finite groups","volume":"92","year":"2019","journal-title":"J. Symb. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"102287","DOI":"10.1016\/j.jsc.2023.102287","article-title":"Fast computation of the centralizer of a permutation group in the symmetric group","volume":"123","year":"2024","journal-title":"J. Symb. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/0012-365X(94)90122-8","article-title":"Isomorphism of some graph coverings","volume":"128","author":"Sato","year":"1994","journal-title":"Discret. Math."},{"key":"ref_28","unstructured":"Venkatesh, A. (1998). Graph Coverings and Group Liftings, Department of Mathematics, University of Western Australia. preprint."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.jctb.2014.09.004","article-title":"2-Arc-transitive metacyclic covers of complete graphs","volume":"111","author":"Xu","year":"2015","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_30","first-page":"269","article-title":"Covers of digraphs","volume":"30","author":"Harary","year":"1980","journal-title":"Math. Slovaca"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Fiala, J., and Seifrtov\u00e1, M. (2024). A novel approach to covers of multigraphs with semiedges. Discuss. Math. Graph Theory, in print.","DOI":"10.7151\/dmgt.2540"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/375\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:16:53Z","timestamp":1760105813000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/3\/375"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,20]]},"references-count":31,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["sym16030375"],"URL":"https:\/\/doi.org\/10.3390\/sym16030375","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,3,20]]}}}