Beyond adjacency: Graph encoding with reachability and shortest paths

Graph-structured data is central to many scientific and industrial domains, where the goal is often to optimize objectives defined over graph structures. Given the combinatorial complexity of graph spaces, such optimization problems are typically addressed using heuristic methods, and it remains unclear how to systematically incorporate structural constraints to effectively reduce the search space. This paper introduces explicit optimization formulations for graph search space that encode properties such as reachability and shortest paths. We provide theoretical guarantees demonstrating the correctness and completeness of our graph encoding. To address the symmetry issues arising from graph isomorphism, we propose lexicographic constraints over neighborhoods to eliminate symmetries and theoretically prove that adding those constraints will not reduce the original graph space. Our graph encoding, along with the corresponding symmetry-breaking constraints, forms the basis for downstream optimization tasks over graph spaces.