Distributed Property Testing for Subgraph-Freeness Revisited

In the subgraph-freeness problem, we are given a constant-size graph H, and wish to determine whether the network contains H as a subgraph or not. The property-testing relaxation of the problem only requires us to distinguish graphs that are H-free from graphs that are -far from H-free, in the sense that an -fraction of their edges must be removed to obtain an H-free graph. Recently, Censor-Hillel et. al. and Fraigniaud et. al. showed that in the property-testing regime it is possible to test H-freeness for any graph H of size 4 in constant time, O(1/ ) rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs H of size 4 cannot test 5-cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that 5-cycle-freeness, and indeed H-freeness for many other graphs H comprising more than 4 vertices, can be tested in constant time. We show that Ck-freeness can be tested in O(1/ ) rounds for any cycle Ck, improving on the running time of O(1/ ) of the previous algorithms for triangle-freeness and C4-freeness. In the special case of triangles, we show that triangle-freeness can be solved in O(1) rounds independently of , when is not too small with respect to the number of nodes and edges. We also show that T -freeness for any constant-size tree T can be tested in O(1) rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in O(1/ ) rounds. This class includes all graphs over 5 vertices except the 5-clique, K5. For cliques Ks over s ≥ 3 nodes, we show that Ks-freeness can be tested in O(m 1/2−1/(s−2)/ 1/2+1/(s−2)) rounds, where m is the number of edges in the graph. Finally, we gives two lower bounds, showing that some dependence on is necessary when testing H-freeness for specific subgraphs H.