Abstract: In a systolic array, the maximum operating speed is determined by the most complex operation performed. In a systolic army graphics engine, capable of generating high quality images, one has to perform complex operations at a very high speed. We propose to use pipelined functional units in systolic army graphics engines as they can perform complex operations at high speeds. Due to time-varying discontinuities of operations performed by systolic army graphics engines, introduction of pipelined functional units is a complex problem. In this paper we present a methodology which solves this problem by a graph-theoretic approach. Furthermore, we characterize the architectures which can be improved by pipelined functional units.