Single point Seshadri constants on rational surfaces

Motivated by a similar result of Dumnicki, Küronya, Maclean and Szemberg under a slightly stronger hypothesis, we exhibit irrational single-point Seshadri constants on a rational surface X obtained by blowing up very general points of PC2, assuming only that all prime divisors on X of negative self-intersection are smooth rational curves C with C2=−1. (This assumption is a consequence of the SHGH Conjecture, but it is weaker than assuming the full conjecture.)