Let G be a K-1,K-r-free graph (r greater than or equal to 3) on n vertices.
We prove that, for any induced path or induced cycle on k vertices in G(k
greater than or equal to 2r-1 or k greater than or equal to 2r, respectivel
y), the degree sum of its vertices is at most (2r - 2)(n - alpha) where ci
is the independence number of G. As a corollary we obtain an upper bound on
the length of a longest induced path and a longest induced cycle in a K-1,
K-r-free graph. Stronger bounds are given in the special case of claw-free
graphs (i.e., r = 3). Sharpness examples are also presented. (C) 2001 John
Wiley & Sons. Inc.