caa a22 4500 475778308 CHVBK 20180406123637.0 cr unu---uuuuu 170329e20000601xx s 000 0 eng 10.1007/PL00021178 doi (NATIONALLICENCE)springer-10.1007/PL00021178 Matroids and Graphs with Few Non-Essential Elements [Elektronische Daten] [James Oxley, Haidong Wu] Abstract.:  An essential element of a 3-connected matroid M is one for which neither the deletion nor the contraction is 3-connected. Tutte's Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3-connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3-connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3-connected matroid M for which no single-element contraction is 3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3-connected. Springer-Verlag Tokyo, 2000 Oxley James Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA. e-mail: oxley@math.lsu.edu, US aut Wu Haidong Department of Mathematics, Southern University, Baton Rouge, LA 70813-9259, USA e-mail: hwu@engr.subr.edu, US aut https://doi.org/10.1007/PL00021178 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/PL00021178 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Oxley James Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918, USA. e-mail: oxley@math.lsu.edu, US aut NATIONALLICENCE 700 1- Wu Haidong Department of Mathematics, Southern University, Baton Rouge, LA 70813-9259, USA e-mail: hwu@engr.subr.edu, US aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer