Fragile Graphs with Small Independent Cuts Guantao Chen, 1 * Ralph J. Faudree, 2 and Michael S. Jacobson 3 1 DEPARTMENT OF MATHEMATICS AND STATISTICS GEORGIA STATE UNIVERSITY ATLANTA, GEORGIA 30303 E-mail: gchen@cs.gsu.edu 2 DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF MEMPHIS MEMPHIS, TENNESSEE 38152 3 DEPARTMENT OF MATHEMATICS UNIVERSITY OF LOUISVILLE LOUISVILLE, KENTUCKY 40292 Received August 28, 2000; Revised June 6, 2002 DOI 10.1002/jgt.10074 Abstract: A graph is called fragile if it has a vertex cut which is also an independent set. Chen and Yu proved that every graph with n vertices and at most 2n  4 edges is fragile, which was conjectured to be true by Caro. However, their proof does not give any information on the number of vertices in the independent cuts. The purpose of this paper is to investigate when a graph has a small independent cut. We show that if G is a graph on —————————————————— This work was partially done during Graph ColoringWorkshop sponsored by Erdo˝s Center in the Summer 1999 at Budapest, Hungary. Contract grant sponsor: NSF; Contract grant number DMS-0070059 (to G. C.). *Correspondence to: Guantao Chen, Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303-3083. E-mail: gchen@cs.gsu.edu ß 2002 Wiley Periodicals, Inc.