The disconnection number of a graph

The disconnection number d ( X ) is the least number of points in a connected topological graph X such that removal of d ( X ) points will disconnect X (Nadler, 1993 [6]). Let D n denote the set of all homeomorphism classes of topological graphs with disconnection number n. The main result characterizes the members of D n + 1 in terms of four possible operations on members of D n . In addition, if X and Y are topological graphs and X is a subspace of Y with no endpoints, then d ( X ) ⩽ d ( Y ) and Y obtains from X with exactly d ( Y ) − d ( X ) operations. Some upper and lower bounds on the size of D n are discussed. gorithm of the main result has been implemented to construct the classes D n for n ⩽ 8 , to estimate the size of D 9 , and to obtain information on certain subclasses such as non-planar graphs ( n ⩽ 9 ) and regular graphs ( n ⩽ 10 ).