A Lower Bound on the Size of Series-Parallel Graphs Dense in Long Paths

One way to quantify how dense a multidag is in long paths is to find the largest n, m such that whichever ≤ n edges are removed, there is still a path from an original input to an original output with ≥ m edges the larger we can make n, m, the denser is the graph. For a given n, m, we would like to lower bound the size such a graph, say in edges, at least when restricting to a particular class of graphs. A bound of Ω(n lgm) was provided in [Val77] for one notion of series-parallel graphs. Here we reprove the same result but in greater detail and relate that notion of series-parallel to other popular notions of series-parallel. In particular, we show that that notion is more general than minimal series-parallel and two terminal series-parallel.