  In earlier work the Kauffman bracket polynomial was extended to an invariant
of marked graphs, i.e., looped graphs whose vertices have been partitioned into
two classes (marked and not marked). The marked-graph bracket polynomial is
readily modified to handle graphs with weighted vertices. We present formulas
that simplify the computation of this weighted bracket for graphs that contain
twin vertices or are constructed using graph composition, and we show that
graph composition corresponds to the construction of a link diagram from
tangles.
