Here’s a really pragmatic use of graph theory. This is a better method of matching compatible kidney donors and recipients than the current inefficient method of arranging transplants.

Here’s the problem. There are 70,000 people on a waitlist for a kidney. Family members may offer to donate kidneys but they are incompatable. Nor do they offer to randomly donate kidneys to absolute strangers.

Could they arrange swaps? For instance, what if a different family was in a similar situation. Someone is willing to donate but they are incompatable. But they are compatible with the other family. So they arrange to swap kidneys. This grows more complicated when you have to arrange intricate swaps between six or seven families to get the compatibility right.

It would save many lives but no one even tries to do it.

(US Naval Academy mathematician Sommer) Gentry expects that if there were a national registry in place for kidney matching, and if it used her team’s method, then each month, about half the pairs in the registry would find a compatible match. Each year, 1,000–2,000 patients would get kidneys who currently would not.

By contrast, as of 2005, only 51 patients had ever received kidneys through a swap in which two incompatible pairs exchange donors to create compatible pairs. Gentry calculates that developing a national registry could save \$750 million per year, because dialysis, the only alternative to transplantation, is very expensive.

Here’s an example of how it works. Draw a ring of circles. Each of these circles are different colors – red, blue, yellow, green. The way it is set up now, the circles (nodes) are only connected to the circle next to it. This means that the travel distance (jumping from linked node to another) is very long.

The graph method would find a way to connect them. Draw lines through the center of your graph – just a couple – to link two nodes together on opposite ends. This reduces the distance to travel to any other node in the graph.

Currently, the red circles need red organs and blue circles need blue organs but there is little linkage in the graph. If a red circle gets the blue organ today, it has to discard it because it is useless. He cannot communicate with blue nodes and share it in a “marketplace” in the center.

With this system, connections can be identifies and made. If a red node gets a blue organ, a blue node gets a green organ, and a green node gets a red organ, instead of disposing the unusable organs, they can organize a complicated organ swap.