# Measuring the distance to series-parallelity by path expressions

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## Abstract

Many graph and network problems are easily solved in the special case of series-parallel networks, but are highly intractable in the general case. This paper considers two complexity measures of *two-terminal directed acyclic graphs* (st-dags) describing the “distance” of an st-dag from series-parallelity. The two complexity measures are the *factoring complexity ψ(G)* and *the reduction complexity μ(G)*. Bein, Kamburowski, and Stallmann [3] have shown that *ψ(G)≤μ(G)≤n*−3, where *G* is an st-dag with *n* nodes. They conjectured that *ψ(G)=μ(G)*. This paper gives a proof for this conjecture.

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© Springer-Verlag Berlin Heidelberg 1995