A Fast Deterministic Detection of Small Pattern Graphs in Graphs Without Large Cliques

Abstract

We show that for several pattern graphs on four vertices (e.g., $C_4$), their induced copies in host graphs with n vertices and no clique on $k+1$ vertices can be deterministically detected in time $\tilde{O}(n^{\omega }k^{\mu }+n^2k^2),$ where $\tilde{O}(f)$ stands for $O(f (\log f)^c )$ for some constant c,  and $\mu \approx 0.46530$. The aforementioned pattern graphs have a pair of non-adjacent vertices whose neighborhoods are equal. By considering dual graphs, in the same asymptotic time, we can also detect four vertex pattern graphs, that have an adjacent pair of vertices with the same neighbors among the remaining vertices (e.g., $K_4$), in host graphs with n vertices and no independent set on $k+1$ vertices.

By using the concept of Ramsey numbers, we can extend our method for induced subgraph isomorphism to include larger pattern graphs having a set of independent vertices with the same neighborhood and n-vertex host graphs without cliques on $k+1$ vertices (as well as the pattern graphs and host graphs dual to the aforementioned ones, respectively).