There are numerous NP-hard combinatorial problems which involve searching for an undirected graph satisfying a certain property. One way to solve such problems is to translate a problem into an instance of the boolean satisfiability (SAT) or constraint satisfaction (CSP) problem. Such reduction usually can give rise to numerous isomorphic representations of the same graph. One way to reduce the search space and speed up the search under these conditions is to introduce symmetrybreaking predicates. In this paper we introduce three novel and practically effective symmetry-breaking predicates for an undirected connected graph search based on breadth-first search (BFS) enumeration and compare with existing symmetry-breaking methods on several graph problems.