Fine-Grained Classification Of Detecting Dominating Patterns

We consider the following generalization of dominating sets: Let $G$ be a host graph and $P$ be a pattern graph $P$. A dominating $P$-pattern in $G$ is a subset $S$ of vertices in $G$ that (1) forms a dominating set in $G$ \emph{and} (2) induces a subgraph isomorphic to $P$. The graph theory literature studies the properties of dominating $P$-patterns for various patterns $P$, including cliques, matchings, independent sets, cycles and paths. Previous work (Kunnemann, Redzic 2024) obtains algorithms and conditional lower bounds for detecting dominating $P$-patterns particularly for $P$ being a $k$-clique, a $k$-independent set and a $k$-matching. Their results give conditionally tight lower bounds if $k$ is sufficiently large (where the bound depends the matrix multiplication exponent $\omega$). We ask: Can we obtain a classification of the fine-grained complexity for \emph{all} patterns $P$? Indeed, we define a graph parameter $\rho(P)$ such that if $\omega=2$, then $$\left(n^{\rho(P)} m^{\frac{|V(P)|-\rho(P)}{2}}\right)^{1\pm o(1)}$$ is the optimal running time assuming the Orthogonal Vectors Hypothesis, for all patterns $P$ except the triangle $K_3$. Here, the host graph $G$ has $n$ vertices and $m=\Theta(n^\alpha)$ edges, where $1\le \alpha \le 2$. The parameter $\rho(P)$ is closely related (but sometimes different) to a parameter $\delta(P) = \max_{S\subseteq V(P)} |S|-|N(S)|$ studied in (Alon 1981) to tightly quantify the maximum number of occurrences of induced subgraphs isomorphic to $P$. Our results stand in contrast to the lack of a full fine-grained classification of detecting an arbitrary (not necessarily \emph{dominating}) induced $P$-pattern.