On lengths of edge-labeled graph expressions

This paper investigates relationship between algebraic expressions and graphs. Our intent is to simplify graph expressions and eventually find their shortest representations. We prove the monotonicity results allowing to assert that the length of a shortest expression of any subgraph of a given graph is not greater than the length of a shortest expression of the graph. We describe the decomposition method for generating expressions of complete st-dags (two-terminal directed acyclic graphs) and estimate the corresponding expression complexities. Using these findings, we present an 2^Olog2n upper bound for the length of a shortest expression for every n-vertex st-dag.