A Comparison of Four Notions of Isomorphism-Based Security for Graphs

A graph is a powerful abstraction for representing information. We address the problem of publishing a secure version of a graph that does not leak information to an adversary who may possess prior information about portions of the graph, and may have unbounded computational power. In this context, we revisit four notions of security, all of which are based on variants of graph isomorphism, that have been proposed in two different application contexts in the literature. We compare the four notions to one another, first from the standpoint of strength, i.e., whether meeting one notion implies meeting another, and then from the standpoint of computational hardness, i.e., what the exact computational complexity is for the problem of checking whether a graph meets a notion. For the latter, we identify that for two of the notions we consider, the problem is $\mathbf{NP}\text{-complete}$ , and for the two others, it is ISO -complete, where ISO is the class of problems induced by graph isomorphism. We observe that strength is not necessarily correlated to computational hardness. In summary, our work makes contributions at the foundations of an important notion of security for graphs.