Ultramassive black holes in the most massive galaxies: $M_{rm BH}-sigma$ versus $M_{rm BH}-R_{rm b}$

[Abridged] We investigate the nature of the relations between black hole (BH) mass ($M_{rm BH}$) and the central velocity dispersion ($sigma$) and, for core-Sersic galaxies, the size of the depleted core ($R_{rm b}$). Our sample of 144 galaxies with dynamically determined $M_{rm BH}$ encompasses 24 core-Sersic galaxies, thought to be products of gas-poor mergers, and reliably identified based on high-resolution HST imaging. For core-Sersic galaxies -- i.e., combining normal-core ($R_{rm b} < 0.5 $ kpc) and large-core galaxies ($R_{rm b} gtrsim 0.5$ kpc), we find that $M_{rm BH}$ correlates remarkably well with $R_{rm b}$ such that $M_{rm BH} propto R_{rm b}^{1.20 pm 0.14}$ (rms scatter in log $M_{rm BH}$ of $Delta_{rm rms} sim 0.29$ dex), confirming previous works on the same galaxies except three new ones. Separating the sample into Sersic, normal-core and large-core galaxies, we find that Sersic and normal-core galaxies jointly define a single log-linear $M_{rm BH}-sigma$ relation $M_{rm BH} propto sigma^{ 4.88 pm 0.29}$ with $Delta_{rm rms} sim 0.47$ dex, however, at the high-mass end large-core galaxies (four with measured $M_{rm BH}$) are offset upward from this relation by ($2.5-4) times sigma_{rm s}$, explaining the previously reported steepening of the $M_{rm BH}-sigma$ relation for massive galaxies. Large-core spheroids have magnitudes $M_{V} le -23.50$ mag, half-light radii Re $>$ 10 kpc and are extremely massive $M_{*} ge 10^{12}M_{odot}$. Furthermore, these spheroids tend to host ultramassive BHs ($M_{rm BH} ge 10^{10}M_{odot}$) tightly connected with their $R_{rm b}$ rather than $sigma$. The less popular $M_{rm BH}-R_{rm b}$ relation exhibits $sim$ 62% less scatter in log $M_{rm BH}$ than the $M_{rm BH}- sigma$ relations.