  Motivated by a similar result of Dumnicki, K\"uronya, 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 $\mathbb{P}^2_\mathbb{C}$, assuming only that all prime
divisors on $X$ of negative self-intersection are smooth rational curves $C$
with $C^2=-1$. (This assumption is a consequence of the SHGH Conjecture, but it
is weaker than assuming the full conjecture.)
