caa a22 4500 605524963 CHVBK 20210128100756.0 cr unu---uuuuu 210128e20150101xx s 000 0 eng 10.1007/s10958-014-2198-z doi (NATIONALLICENCE)springer-10.1007/s10958-014-2198-z Karpov D. St.Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia aut The Decomposition Tree of a Biconnected Graph [Elektronische Daten] [D. Karpov] The decomposition tree of a biconnected graph is in brief the decomposition tree of a biconnected graph by the set of all single cutsets of it (i.e., 2-vertex cutsets that are independent with all other 2-vertex cutsets). It is shown that this tree has much in common with the classical tree of blocks and cutpoints of a connected graph. With the help of the decomposition tree of a biconnected graph, a planarity criterion is proved and some upper bounds on the chromatic number of this graph are found. Finally, the structure of critical biconnected graphs is studied, and it is proved that each such graph has at least four vertices of degree 2. Bibliography: 11 titles. Springer Science+Business Media New York, 2014 Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 204/2(2015-01-01), 232-243 1072-3374 204:2<232 2015 204 10958 https://doi.org/10.1007/s10958-014-2198-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-014-2198-z text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Karpov D. St.Petersburg Department of the Steklov Mathematical Institute, St. Petersburg State University, St. Petersburg, Russia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 204/2(2015-01-01), 232-243 1072-3374 204:2<232 2015 204 10958