🤖 AI Summary
This paper investigates the *edge-induced density problem*—a unifying refinement of both the Kruskal–Katona theorem and the Pippenger–Golumbic induced density problem. To bridge local structure and global density, we introduce the novel framework of *local directed graphs*, establishing for the first time a fundamental connection between edge-induced and vertex-induced densities. Combining extremal graph theory, fractional matching theory, and constructive graph transformations, we prove NP-hardness of the problem on general graphs and obtain an exact characterization for graphs admitting a unique fractional perfect matching. We fully determine the edge-induced densities of all graphs on at most four vertices, improve boundary results for $C_5$ and $P_6$, and uncover deep links between structural properties—such as matching uniqueness and symmetry—and computational complexity. This work provides a new paradigm for induced density theory.
📝 Abstract
In this paper we introduce the edge inducibility problem. This is a common refinement of both the well known Kruskal--Katona theorem and the inducibility question introduced by Pippenger and Golumbic.
Our first result is a hardness result. It shows that for any graph $G$, there is a related graph $G'$ whose edge inducibility determines the vertex inducibility of $G$. Moreover, we determine the edge inducibility of every $G$ with at most $4$ vertices, and make some progress on the cases $G=C_5,P_6$. Lastly, we extend our hardness result to graphs with a perfect matching that is the unique fractional perfect matching. This is done by introducing locally directed graphs, which are natural generalizations of directed graphs.
