Reducibility of Secular Polynomials

Analysing graphs on up to three edges we noted that the secular polynomials are reducible if and only if the graphs either contain loops or are watermelon graphs. It is our goal in this chapter to prove this statement for arbitrary finite graphs (Theorem 7.19). We are going to use contractions and extensions of graphs allowing us to consider just a few specially chosen families of graphs together with graphs on up to three edges. These families are studied as a preparation for the proof of the factorisation theorem.