On Algebraic Expressions of Series-Parallel and Fibonacci Graphs

The paper investigates relationship between algebraic expressions and graphs. Through out the paper we consider two kinds of digraphs: series-parallel graphs and Fibonacci graphs (which give a generic example of non-series-parallel graphs). Motivated by the fact that the most compact expressions of series-parallel graphs are read-once formulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n) length for Fibonacci graphs. A serious e¤ort was made to prove that this algorithm yields expressions with a minimum number of terms. Using an interpretation of a shortest path algorithm as an algebraic expression, a symbolic approach to the shortest path problem is proposed.