That doesn't seem to be the case all the time. What about command
posts?
Ya its defninitely not the case. I just attacked a planet that had a sentry eye on it, with the intention of neutering it but not killing the command station. Due to the eye, I had to kill every Command Post in sequence, slowly, and it gave me ample time to see exactly what caused the two neighboring planets to go on alert. It wasn't killing the command posts, and it wasn't the death of the eye. It happened right when I killed the warp gate.
Maybe it was the overwhelming numbers I brought to kill the warp gate? Does that have an effect? Prior to the death of the eye I only had about 200 ships max on the planet, but when the eye went down i sent in about 600 to push for the warp gate. Either way, the planets next door went on alert only after I killed the gate.