For starters, “*Graph Theory*” has nothing to do with drawing or creating your atypical graph as in a pie chart or bar graph. Instead this is the study of “*graphs*” as in mathematical structures used to represent and model pair-wise relations between objects from a given collection or in our case “*set*” (remember from yesterdays failure set). As here we will refer to a “*graph*” in this context of ‘nodes’ and the set of edges which connect pairs of nodes.

In fact, “*graphs*” are among the most ubiquitous models of both natural and human-made structures as they can be used to model many types of relations and process dynamics in various systems. Many problems of practical interest can be represented by graphs and this makes them especially well suited for problem solving and to demonstrate lets use a simple example of a gasoline engine.

As here we need three things to make the internal combustion motor run and a lack of any one will cause the engine to fail. As we see from the graphic the key components are Spark, Air and Fuel, while each are backed by a series of sub-components. So you happen to be a mechanic and a client brings you a non-running car, here you can simply step through each node to find the failure point. So in short these nodes represent our failure set and a rather basic one at that, as each path is pretty much self contained and being such form a sub-failure set itself.

Again this is a simple example and walking each step doesn’t yield that much time pressure of course unless you’re on the side of a freeway on your way to work to meet a large client. Now we have a different story so how do we shorten the path? In academic terms there are many methods like Dijkstra, Eulerian Circuit, Hamilton Circuit and so on down the line. However for our road side emergency we will apply simple deductive reasoning.

Thus the simplest path to rule out is “*Air*” as a quick inspection will show the status as in blocked or unblocked. We then seek the next “*quickest*” path to rule out that there is spark as we can pull a plug wire and see if it arch’s to ground, if yes the only remaining sub-component failure path is the fuel system.

It’s also worth noting that this is how your GPS finds its way as it creates a “*graph*” of your potential trip, treating all the roadways as nodes and then calculates the path based upon the best “*costing*” of the route. In the GPS example a “*Cost*” could be “*speed*” as a slower road would have a “*higher cost*” then say a freeway. If you have an iOS device, a nice application to see how this works is “*GraphSolver for iPhone*” as it will also run on the iTouch and iPad too and provide a visual display of the graph and solution.

So the next time you have a problem to solve, start out by drawing a “*graph map*” of the failure sets and mess around with it a bit as you will be amazed. As just like Napoleon Bonaparte said, “*a picture is worth a thousand words*”, whereas I say a graph map is worth a thousand aspirin…