Chromatic Polynomials of Signed Graphs and Dominating-Vertex Deletion Formulae

We exhibit non-switching-isomorphic signed graphs that share a common underlying graph and common chromatic polynomials, thereby answering a question posed by Zaslavsky. We introduce a new pair of bivariate chromatic polynomials that generalises the chromatic polynomials of signed graphs. We establish recursive dominating-vertex deletion formulae for these bivariate chromatic polynomials. As an application, we demonstrate that for a certain family of signed threshold graphs, isomorphism can be characterised by the equality of bivariate chromatic polynomials.