Disconnection Numbers, Pizza Cuts, and Cycle Rank

In this paper we bring out more strongly the connection between the disconnection number of a graph and its cycle rank. We also show how to associate with a pizza sliced right across in a certain way with \(n-2\) cuts a graph with \(n\) vertices, and show that if the pizza is cut thereby into \(r\) pieces, then any set of \(r-1\) of these pieces corresponds to a basis for the cycle space of the associated graph. Finally we use this to explain why for \(n\geq 3\) the greatest number of regions that can be formed by slicing a pizza in the certain way with \(n-2\) cuts, namely \(\frac{1}{2}(n^2-3n+4)\), equals the disconnection number of \(K_n\).