In this paper we introduce an adaptive technique for compressing small quantities of text which are organized as a rooted directed graph. We impose a constraint on the technique such that data encountered during a traversal of any valid path through the graph must be recoverable without requiring the expansion of data that is not on the path in question. The technique we present determines the set of nodes y which are guaranteed to be encountered before reaching node x while traversing any valid path in the graph, and uses them as a basis for conditioning an LZW dictionary for the compression/expansion of the data in x. Initial results show that our improved LZW technique reduces the compressed text size by approximately 20% more than regular LZW, and requires only minor modifications to the standard LZW decompression routine.