Abstract
We define a new graph polynomial, the polynomial, for any undirected graph. Also, we show how to count Euler circuits and circuitdecompositions for any directed or undirected Eulerian graph, by a straightforward reduction formula. For 2-in, 2-out directed graphs D, any Euler circuit induces an undirected interlace graph H, and there is a close relationship between the number of circuit decompositions of D and the polynomial of H. We explore this relationship, and properties of the polynomial in general.
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