Menger's Paths with Minimum Mergings

TL;DR

This paper establishes bounds on the number of mergings in Menger's paths within acyclic directed graphs, linking these bounds to min-cuts between sources and sinks, regardless of graph size or topology.

Contribution

It proves the existence of Menger's paths with a bounded number of mergings depending only on min-cuts, providing new theoretical bounds and insights.

Findings

Number of mergings is bounded by a constant related to min-cuts.

Bounds on minimum mergings depend on min-cuts.

Results hold for any acyclic directed graph with multiple sources and sinks.

Abstract

For an acyclic directed graph with multiple sources and multiple sinks, we prove that one can choose the Merger's paths between the sources and the sinks such that the number of mergings between these paths is upper bounded by a constant depending only on the min-cuts between the sources and the sinks, regardless of the size and topology of the graph. We also give bounds on the minimum number of mergings between these paths, and discuss how it depends on the min-cuts.