On Algebraic Expressions of Series-Parallel and Fibonacci Graphs Mark Korenblit and Vadim E. Levit Department of Computer Science Holon Academic Institute of Technology Holon, Israel {korenblit, levitv}@hait.ac.il Abstract. The paper investigates relationship between algebraic ex- pressions 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 for- mulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n 2 ) 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. 1 Introduction A graph G =(V;E) consists of a vertex set V and an edge set E, where each edge corresponds to a pair (v;w) of vertices. If the edges are ordered pairs of vertices (i.e., the pair (v;w) is di/erent from the pair (w; v)), then we call the graph directed or digraph ; otherwise, we call it undirected.A path from vertex v 0 to vertex v k in a graph G =(V;E) is a sequence of its vertices v 0 ;v 1 ;v 2 ; :::; v k1 ;v k , such that (v i1 ;v i ) 2 E for 1  i  k. A graph G 0 =(V 0 ;E 0 ) is a subgraph of G =(V;E) if V 0  V and E 0  E. A graph G is homeomorphic to a graph G 0 (homeomorph of G 0 ) if G can be obtained by subdividing edges of G 0 via adding new vertices. A two-terminal directed acyclic graph (st-dag ) has only one source s and only one sink t. In an st-dag, every vertex lies on some path from the source to the sink. An algebraic expression is called an st-dag expression if it is algebraically equivalent to the sum of products corresponding to all possible paths between the source and the sink of the st-dag [1]. This expression consists of terms (edge labels) and the operators + (disjoint union) and  (concatenation, also denoted by juxtaposition when no ambiguity arises). We dene the total number of terms in an algebraic expression, including all their appearances, as the complexity of the algebraic expression. The complexity of an st-dag expression is denoted by T (n), where n is the number of vertices in