Deletion–contraction formula

In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form:

${\displaystyle f(G)=f(G\setminus e)+f(G/e).}$

Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such a function as a W-function. The formula is sometimes referred to as the fundamental reduction theorem. In this article we abbreviate to DC.

R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more, including a function f = t(G) counting the number of spanning trees of a graph (also see Kirchhoff's theorem). It was later found that the flow polynomial is yet another; and soon Tutte discovered an entire class of functions called Tutte polynomials (originally referred to as dichromates) that satisfy DC.