Lemos, Manoel; Oxley, James On removable cycles through every edge. (English) Zbl 1009.05083 J. Graph Theory 42, No. 2, 155-164 (2003). Summary: W. Mader [Abh. Math. Semin. Univ. Hamb. 42, 187-204 (1974; Zbl 0266.05111)] and B. Jackson [J. Lond. Math. Soc., II. Ser. 21, 385-392 (1980; Zbl 0438.05038)] independently proved that every 2-connected simple graph \(G\) with minimum degree at least four has a removable cycle, that is, a cycle \(C\) such that \(G\setminus E(C)\) is 2-connected. This paper considers the problem of determining when every edge of a 2-connected graph \(G\), simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from \(G\) remains 2-connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge-disjoint removable cycles such that each contains one of the distinguished edges. Cited in 3 Documents MSC: 05C38 Paths and cycles 05B35 Combinatorial aspects of matroids and geometric lattices Keywords:removable cycle; 2-connected graph; matroid Citations:Zbl 0266.05111; Zbl 0438.05038 PDFBibTeX XMLCite \textit{M. Lemos} and \textit{J. Oxley}, J. Graph Theory 42, No. 2, 155--164 (2003; Zbl 1009.05083) Full Text: DOI References: [1] and Graph Theory with Applications, North-Holland, New York, 1976. · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2 [2] Fleischner, J London Math Soc 31 pp 193– (1985) [3] Goddyn, J Combin Theory Ser B 71 pp 130– (1997) [4] Goddyn, Combin Probab Comput 8 pp 539– (1999) [5] Jackson, J London Math Soc 21 pp 385– (1980) [6] Lemos, J Graph Theory 30 pp 51– (1999) [7] Mader, Abh Math Sem Univ Hamburg 42 pp 187– (1974) [8] Matroid Theory, Oxford University Press, New York, 1992. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.