{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T10:15:19Z","timestamp":1773656119222,"version":"3.50.1"},"reference-count":13,"publisher":"Wiley","issue":"7","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":5062,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1992,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We describe an algorithm for computing the number of <jats:italic>k<\/jats:italic>\u2010component spanning forests of a graph <jats:italic>G<\/jats:italic> that runs in polynomial time for fixed <jats:italic>k<\/jats:italic>. The algorithm is based on earlier work by Liu and Chow. Our contributions are a simpler graph\u2010theoretic proof of their formula and a demonstration of how Jacobi's Theorem can be applied to improve the asymptotic time complexity. By matroid duality, the number of connected spanning subgraphs of cyclomatic number <jats:italic>c<\/jats:italic> of a planar graph equals the number of <jats:italic>c<\/jats:italic> + 1\u2010component forests in the dual. Thus, one application of this research is an algorithm for counting connected spanning unicyclic subgraphs of a planar graph. We show this can be done in time <jats:italic>O(M(n))<\/jats:italic>, where <jats:italic>M(n)<\/jats:italic> is the time complexity of multiplying together two <jats:italic>n<\/jats:italic> by <jats:italic>n<\/jats:italic> matrices.<\/jats:p>","DOI":"10.1002\/net.3230220704","type":"journal-article","created":{"date-parts":[[2007,5,12]],"date-time":"2007-05-12T13:21:17Z","timestamp":1178976077000},"page":"647-652","source":"Crossref","is-referenced-by-count":8,"title":["Counting <i>k<\/i>\u2010component forests of a graph"],"prefix":"10.1002","volume":"22","author":[{"given":"Wendy","family":"Myrvold","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"The Design and Analysis of Computer Algorithms","author":"Aho A. V.","year":"1974"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-349-03521-2"},{"key":"e_1_2_1_4_2","series-title":"The International Series of Monographs on Computer Science","volume-title":"The Combinatorics of Network Reliability","author":"Colbourn C. J.","year":"1987"},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","unstructured":"D.CoppersmithandS.Winograd Matrix multiplications via arithmetic progressions.19th Annual STOC(1987)1\u20136.","DOI":"10.1145\/28395.28396"},{"key":"e_1_2_1_6_2","volume-title":"Algorithmic Graph Theory","author":"Gibbons A.","year":"1985"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1145\/321850.321852"},{"key":"e_1_2_1_8_2","volume-title":"The Theory of Matrices in Numerical Analysis","author":"Householder A. S.","year":"1964"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.2307\/2044142"},{"key":"e_1_2_1_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01994058"},{"key":"e_1_2_1_11_2","first-page":"261","article-title":"Bounds for all\u2010terminal reliability in planar networks","volume":"33","author":"Ramesh A.","year":"1987","journal-title":"Ann. Discrete Math."},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190080304"},{"key":"e_1_2_1_13_2","volume-title":"Linear Algebra and Its Applications","author":"Strang G.","year":"1976"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165411"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230220704","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230220704","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T09:40:14Z","timestamp":1698054014000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230220704"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1992,12]]},"references-count":13,"journal-issue":{"issue":"7","published-print":{"date-parts":[[1992,12]]}},"alternative-id":["10.1002\/net.3230220704"],"URL":"https:\/\/doi.org\/10.1002\/net.3230220704","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1992,12]]}}}