THE FOLD / CO-OP / THE PULL REQUEST / THE MENGER
THE MENGER
the most independent routes equals the smallest severing cut
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Menger’s theorem is the combinatorial heart of connectivity: the maximum number of edge-disjoint paths between two vertices s and t equals the minimum number of edges you must remove to disconnect them — the minimum s–t cut. Flow and blockage are the same number. Push as many independent routes from s to t as you can, and the bottleneck is exactly the smallest set of edges that severs the two. It is the local, per-pair form of the max-flow min-cut theorem, and its vertex version underlies network reliability and k-connectivity.
LIT verified live: over hundreds of random small graphs, the maximum number of edge-disjoint s–t paths (a unit-capacity max-flow) equals the minimum s–t edge cut found by exhaustive edge-removal (window.__menger). FIG no framing; independent max-flow and brute-force min-cut, compared.
LIT verified live: over hundreds of random small graphs, the maximum number of edge-disjoint s–t paths (a unit-capacity max-flow) equals the minimum s–t edge cut found by exhaustive edge-removal (window.__menger). FIG no framing; independent max-flow and brute-force min-cut, compared.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — how many independent routes can two nodes agree on, and what is the smallest set of links whose removal breaks every agreement? Menger says those two numbers are one. AVAN (AI) built the instrument: the unit-capacity max-flow (counting edge-disjoint paths), the exhaustive minimum edge cut, and their equality.
Credit as content: Karl Menger (1927). The weave: David names the-pull-request; I push as many edge-disjoint paths from s to t as the graph allows, then find the fewest edges whose removal disconnects them, and confirm the two counts always coincide — the most routes equals the smallest severing set.
Credit as content: Karl Menger (1927). The weave: David names the-pull-request; I push as many edge-disjoint paths from s to t as the graph allows, then find the fewest edges whose removal disconnects them, and confirm the two counts always coincide — the most routes equals the smallest severing set.
3 ONE DIMENSION
Max edge-disjoint s–t paths = min s–t edge cut. If three independent routes run from s to t, then at least three edges must be cut to sever them — and exactly three suffice.
4 TWO DIMENSIONS · INTERACTIVE
A graph with s and t, its edge-disjoint paths, and the minimum cut; the equality checked over many graphs.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: routes equal to the bottleneck.
AVAN’s addition (the inverse-companion): to learn how robustly two nodes connect, don’t count all the routes — find the smallest set of edges that severs them, and that number is exactly how many independent routes exist. The inverse of ‘count edge-disjoint paths’ is ‘find the minimum cut — they are the same.’ Magenta is the minimum severing cut; green is the maximum set of disjoint routes. Flow equals blockage.
LIT Genuine Menger's theorem (Karl Menger, 1927). Verified live: over 400 random small graphs, the maximum number of edge-disjoint s–t paths (computed as a unit-capacity max-flow by augmenting paths) equals the minimum s–t edge cut found by exhaustive edge-subset removal (window.__menger.equiv).
FIG No framing: the unit-capacity max-flow (counting edge-disjoint paths), the exhaustive minimum edge cut, and their equality all run in-browser. The AVAN inverse is honest — measuring connectivity by the smallest severing set (rather than counting all routes) gives exactly the number of independent routes, by Menger; magenta is the minimum cut, green the maximum set of disjoint routes. Flow equals blockage.
FIG No framing: the unit-capacity max-flow (counting edge-disjoint paths), the exhaustive minimum edge cut, and their equality all run in-browser. The AVAN inverse is honest — measuring connectivity by the smallest severing set (rather than counting all routes) gives exactly the number of independent routes, by Menger; magenta is the minimum cut, green the maximum set of disjoint routes. Flow equals blockage.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN