{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:52:41Z","timestamp":1753894361523,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T00:00:00Z","timestamp":1667433600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:p>A mixed graph is a set of vertices together with an edge set and an arc set.\nAn $(m,n)$-mixed graph $G$ is a mixed graph whose edges are each assigned one\nof $m$ colours, and whose arcs are each assigned one of $n$ colours. A\n\\emph{switch} at a vertex $v$ of $G$ permutes the edge colours, the arc\ncolours, and the arc directions of edges and arcs incident with $v$. The group\nof all allowed switches is $\\Gamma$.\n  Let $k \\geq 1$ be a fixed integer and $\\Gamma$ a fixed permutation group. We\nconsider the problem that takes as input an $(m,n)$-mixed graph $G$ and asks if\nthere a sequence of switches at vertices of $G$ with respect to $\\Gamma$ so\nthat the resulting $(m,n)$-mixed graph admits a homomorphism to an\n$(m,n)$-mixed graph on $k$ vertices. Our main result establishes this problem\ncan be solved in polynomial time for $k \\leq 2$, and is NP-hard for $k \\geq 3$.\nThis provides a step towards a general dichotomy theorem for the\n$\\Gamma$-switchable homomorphism decision problem.<\/jats:p>","DOI":"10.46298\/dmtcs.9242","type":"journal-article","created":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T13:28:40Z","timestamp":1667482120000},"source":"Crossref","is-referenced-by-count":1,"title":["The 2-colouring problem for $(m,n)$-mixed graphs with switching is polynomial"],"prefix":"10.46298","volume":"vol. 24, no 2","author":[{"given":"Richard C","family":"Brewster","sequence":"first","affiliation":[]},{"given":"Arnott","family":"Kidner","sequence":"additional","affiliation":[]},{"given":"Gary","family":"MacGillivray","sequence":"additional","affiliation":[]}],"member":"25203","published-online":{"date-parts":[[2022,11,3]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/dmtcs.episciences.org\/10217\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/dmtcs.episciences.org\/10217\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T20:32:00Z","timestamp":1687293120000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/dmtcs.episciences.org\/9242"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,3]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2022,11,3]]}},"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.46298\/dmtcs.9242","relation":{"has-preprint":[{"id-type":"arxiv","id":"2203.08070v2","asserted-by":"subject"},{"id-type":"arxiv","id":"2203.08070v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"2203.08070","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.2203.08070","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2022,11,3]]},"article-number":"9242"}}