Disconnection probability of graph on two dimensional manifold

In this paper an improved asymptotic formula for a disconnection probability of a graph placed on a two dimensional surface is obtained. A deduction of this formula is based on concepts of the Euler characteristic, a plane evolvent of the surface, a fundamental group of two dimensional manifold and etc. 1252 G. Sh. Tsitsiashvili, M. A. Osipova, A. S. Losev and Yu. N. Kharchenko Mathematics Subject Classification: 60K10, 94C05