| Summary: | We introduce a new equivalence on graphs, defined by its symmetry-breaking capability. We first present a framework for various backtracking search algorithms, in which the equivalence is used to prune the search tree. Subsequently, we define the equivalence and an optimization problem with the goal of finding an equivalence partition with the highest pruning potential. We also position the optimization problem into the computational-complexity hierarchy. In particular, we show that the verifier lies between <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">P</mi> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">NP</mi> </semantics> </math> </inline-formula>-complete problems. Striving for a practical usability of the approach, we devise a heuristic method for general graphs and optimal algorithms for trees and cycles.
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