He misunderstood the circuits problem. It's not a graph planarity problem - anything significant is not going to be planar. Only very simple circuits are wireable with less than two layers (and then by, for instance, sneaking a couple of lines under a resistor, etc. - not planar).
But you do get to use graphs to solve the component placement problem, in heuristics that try to minimize total wire length. That hopefully means that wiring delays are smaller, and routing more feasible. One of the classic heuristics involved successive bipartions on the min-cut of the graph.
If this is the same Dover book we used in our graph theory class, I found it difficult to learn from without a professor explaining the unclear sections. Not just doing the exercises, but some of the things they presented as trivial had some important steps missing.
But you do get to use graphs to solve the component placement problem, in heuristics that try to minimize total wire length. That hopefully means that wiring delays are smaller, and routing more feasible. One of the classic heuristics involved successive bipartions on the min-cut of the graph.
Nitpicking aside, it's a good post, well argued. My intro book recommendation: http://www.reddit.com/r/mathbooks/comments/aho2h/graph_theor...