Various systems are provided for optimizing the searching of a graph for a portion that matches a pattern is provided. A Graph Search Optimization System (“GSOS”) provides various techniques for reducing the computational expense when searching for patterns within a graph. The GSOS provides techniques that include an edge-count directed (“ECD”) system, a derived constraint (“DC”) system, and a sorted property (“SP”) system. The ECD system matches a pattern in a direction based on the number of edges for that direction. The DC system derives a single-element constraint from a multi-element constraints to avoid having to check multiple elements. The SP system processes edges of a graph in a sorted order based on the value of a property of the edges.