On a conjecture of Thomassen and Toft

This article is motivated by a conjecture of Thomassen and Toft on the numb er s(2)(G) of separating vertex sets of cardinality 2 and the number v(2)(G ) of vertices of degree in a graph G belonging to the class G of all 2-conn ected graphs without nonseparating induced cycles. Let parallel to G parall el to denote the number of edges of the graph G. Thomassen and Toft conject ured in [C. Thomassen & B. Toft, J. Combin; Theory B 31 (1981), 199-224] th e existence of a positive constant c satisfying s(2)(G) + v(2)(G) > c. para llel to G parallel to for all G is an element of G. We shall see that this is nor true in general, Restricting ourselves to planar graphs, we obtain s (2)(G) + v(2)(G) > 1/5 parallel to G parallel to for all planar G is an ele ment of G, where 1/5 is best-possible, (C) 1999 John Wiley Be Sons, Inc. J Graph Theory 32: 118-122, 1999.