Some problems in the enumeration of labelled graphs

A method of counting labelled 3-connected graphs is presented. The enumeration is accomplished by employing a new characterisation of 3-connected graphs, based on an examination of the graphs which may result when a line is removed from a 3-connected graph. The same general principle is first applied to counting connected and 2-connected labelled graphs. In all three cases, it is necessary to keep track of both the numbers of points and of lines. Numerical results are supplied, showing the numbers of connected, 2-connected and 3-connected labelled graphs with p points for p up to 20 .