Many complex combinatorial problems arising from a range of scientific applications (such as computer networks, mathematical chemistry and bioinformatics) involve searching for an undirected graph satisfying a given property. Since for any possible solution there can be a large number of isomorphic representations, these problems can quickly become intractable. One way to mitigate this problem is to eliminate as many isomorphic copies as possible by breaking symmetry during search - i.e. by introducing constraints that ensure that at least one representative graph is generated for each equivalence class, but not the entire class. The goal is to generate as few members of each class as possible - ideally exactly one: the symmetry break is said to be complete in this case. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory. We also illustrate the application of our approach to graph edge coloring problems which exhibit additional symmetries due to the fact that the colors of the edges in any solution can be permuted.

中文翻译：

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图表示中对称破坏的约束

由一系列科学应用（例如计算机网络，数学化学和生物信息学）引起的许多复杂的组合问题都涉及搜索满足给定属性的无向图。因为对于任何可能的解决方案而言，可能存在大量的同构表示，所以这些问题很快就会变得棘手。缓解此问题的一种方法是通过在搜索过程中打破对称性来消除尽可能多的同构副本，即通过引入约束来确保为每个等效类而不是整个类生成至少一个代表性图。目标是在每个类中生成尽可能少的成员-理想情况下恰好一个成员：对称中断据说是完整的在这种情况下。在本文中，我们为无向图搜索引入了新颖，有效和紧凑的对称破缺约束。虽然不完整，但事实证明这些对修剪图形搜索非常有帮助。我们说明了对称表示在图表示中的应用，以解决极值图理论中的几个开放实例。我们还说明了我们的方法在图形边缘着色问题上的应用，该问题由于在任何解决方案中边缘的颜色都可以置换的事实而呈现出额外的对称性。