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 (n lg m) 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.

• Type: Journal
• Year: 2008
• Where: ECCC
• Authors: Chris Calabro
• Keywords: ECCC 2008, Long Paths, Original Input, Original Output