  We introduce a class of pairs of graphs consisting of two cliques joined by
an arbitrary number of edges. The members of a pair have the property that the
clique-bridging edge-set of one graph is the complement of that of the other.
We prove a precise relation between the chromatic polynomials of the graphs in
such a pair, showing that they have the same splitting field, and that the
number of acyclic orientations of each graph is determined by the number of
proper vertex-colourings of the other.
