I may be repeating some of what Larry said; I wrote this offline before seeing his post. But ...
By definition, a perfectly non-compressible rod cannot be made shorter; if it is also perfectly rigid, then it cannot bend. The two ends of such a rod would always be the same distance apart. So a push at one end would have to accelerate all parts of the rod at the instant when a force is applied at one end. If such a rod existed, people at opposite ends of the rod could communicate with one another instantaneously, without having to wait for light to travel for four years each way. Have you figured it out, yet? Not only does such a rod not exist, but its existence would contradict many of the things we think we know about the physical universe. This sort of mental exercises is fun, and its failure to make sense can lead you to a deeper understanding of the universe; but do keep in mind that it is not science but philosophy. There are no irresistible forces, no immovable objects and no non-compressible materials—except in your imagination.
Both tension and compression are transmitted, at the material’s speed of sound, by electromagnetic forces between the electrons of adjacent atoms; electromagnetic forces cannot propagate faster than light. Tension and compression propagate much slower than light because they do not follow a straight line; they zigzag thru the molecular structure.
If you want a plausible scientific problem, consider pulling on in a thin carbon fiber—about as close as you can get to a non-stretchable material—at least that would not absorb the force by bending. So how long would it take for the other end to feel the pull? In fact, the pull would produce a wave of stretching which would propagate at whatever the speed of sound is in the carbon fiber. I’d wild-guess that is about 30,000 meter/sec or 1/10,000 of the speed of light. So it would take about 40,000 years for the pulse reach the other end of a 4 ly-long string.
