Measuring the distance to series-parallelity by path expressions

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.