Solving a variety of physical problems often requires finding the optimal path. One example is using a navigation app to find the best driving route. This is a type of optimization problem where the goal is to find the shortest or weighted shortest (considering factors such as traffic and speed limits) path from a source to a destination in a well-defined network of roads or simply a graph (i.e., a set of nodes and edges) of nodes and edges. However, for the problem of determining the path of a growing crack through a heterogeneous microstructure of a ductile material, both the graph and target may be unknown beforehand. Herein, we propose a simple method that uses evolving graphs and microstructural unit events to infer the crack path and crack growth resistance of two types of material microstructures. The first type consists of a connected network of interfaces (such as a grain boundary network), along which the crack grows. In this case, the interface junctions act as nodes and the interfaces are edges in the graph. The second type involves distributions of discrete second-phase particles, where the particles are nodes but the connections, or edges, between these nodes are unknown beforehand. In both type of microstructures, the target of the growing crack is always unknown beforehand. The detailed crack path and crack growth resistance predictions of this simple method for both types of microstructures are compared with those of the full-field microstructure-based finite deformation finite element calculations of ductile fracture. The two methods show a remarkably good correlation. The fully validated method is then used to computationally design material microstructures with enhanced crack growth resistance.