My parents are retired out in Arizona, and my dad works as a volunteer tour guide at the Titan II Missile Museum near Tucson. (The only Titan II silo still in existence -- though the missile now has no warheads, rocket fuel, or engines, of course! -- since the other 53 sites were dynamited after being decommissioned in the late '80s.)
Anyway, the Titan II missiles and the underground silos that housed them were both designed to deal with that "35 minute" problem described in the graphic. For instance, the missiles used a special fuel mixture that did not have to be stored in huge refrigeration units (like liquid oxygen) and pumped into the rocket's fuel tanks at the last minute -- so the missiles were completely fueled at all times, and could launch in just under 1 minute.
And the silos were massively engineered and theoretically capable of surviving a "nearby" detonation by a Soviet first strike -- not only were there steel doors and concrete walls more than a meter thick, but absolutely everything inside the silo was mounted on gigantic shock-absorption springs to protect electronic equipment from shockwaves. (Thus, even if a multi-megaton Soviet missile had destroyed the nearby city of Tucson, the silo that's now the Museum should have been able -- at least in theory -- to launch a retaliatory strike.)
Thus, the Titan II program was an attempt to preserve the US Govt's official "no first strike" policy, even in an age where Soviet missiles could reach the US in about half an hour. (Note that the magazine article posted by rockzmom is from 1957, and the Titan II silos were completed and activated by around 1963.)