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One of the demanding frontiers in ultracold quantum science is identifying laser cooling schemes for complex atoms and molecules out of their vast spectra of internal states. Motivated by the prospect of expanding the set of available ultracold molecules for applications in fundamental physics, chemistry, astrochemistry, and quantum simulation, we propose and demonstrate an automated graph-based search approach for viable laser cooling schemes. The method is time efficient, reproduces the results of previous manual searches, and reveals a plethora of new potential laser cooling schemes. We discover laser cooling schemes for YO, ${\mathrm{C}}_2, {\mathrm{OH}}^{+}$, CN, and ${\mathrm{CO}}_2$, including surprising schemes that start from highly excited states or do not rely on a strong main transition. A central insight of this work is that the reinterpretation of quantum states and transitions between them as a graph can dramatically enhance the ability to identify new quantum control schemes for complex quantum systems. As such, this approach will also apply to complex atoms and, in fact, any complex many-body quantum system with a discrete spectrum of internal states.