The search for exactly solvable models is an evergreen topic in theoretical physics. In the context of multistate Landau-Zener models -- $N$-state quantum systems with linearly time-dependent Hamiltonians -- the theory of integrability provides a framework for identifying new solvable cases. In particular, it was proved that the integrability of a specific class known as the multitime Landau-Zener (MTLZ) models guarantees their exact solvability. A key finding was that an $N$-state MTLZ model can be represented by data defined on an $N$-vertex graph. While known host graphs for MTLZ models include hypercubes, fans, and their Cartesian products, no other families have been discovered, leading to the conjecture that these are the only possibilities. In this work, we conduct a systematic graph-theoretical search for integrable models within the MTLZ class. By first identifying minimal structures that a graph must contain to host an MTLZ model, we formulate an efficient algorithm to systematically search for candidate graphs for MTLZ models. Implementing this algorithm using computational software, we enumerate all candidate graphs with up to $N = 13$ vertices and perform an in-depth analysis of those with $N \le 11$. Our results corroborate the aforementioned conjecture for graphs up to $11$ vertices. For even larger graphs, we propose a specific family, termed descendants of ``$(0,2)$-graphs'', as promising candidates that may violate the conjecture above. Our work can serve as a guideline to identify new exactly solvable multistate Landau-Zener models in the future.