Edge Inducibility in Graph Theory

• Edge inducibility is a graph theory measure that quantifies the maximum asymptotic density of induced copies of a fixed graph among all m-edge host graphs.
• The topic integrates hardness reductions, explicit extremal constructions for small graphs, and novel locally directed graph techniques to bridge edge and vertex inducibility.
• It has significant implications for combinatorial algorithms and extremal graph theory, offering insights via entropy methods and automorphism-based normalization.

The edge inducibility problem is a broad refinement of classical extremal enumerative results in graph theory, merging perspectives from Kruskal–Katona-style clique enumeration and the asymptotic induced subgraph density questions posed by Pippenger and Golumbic. Its central inquiry is: for a fixed graph $G$ (typically with no isolated vertices), what is the maximum possible asymptotic density of induced copies of $G$ among all graphs with a given number of edges $m$? This density, denoted as $\eind(G, m)$, quantifies the prevalence of $G$ as an induced subgraph in the most “inducible” $m$-edge host. The topic incorporates hardness reductions, explicit extremal constructions for small graphs, new techniques involving locally directed graphs, and a deep connection to vertex inducibility and fractional combinatorial parameters.

1. Edge-Inducibility Definitions and Formulations

For a given graph $G$ (without isolated vertices), the edge inducibility function is defined as: $\eind(G, m) := N_{\ind}(G, H),$ where $H$ is the extremal $m$-edge host graph maximizing the number of induced copies of $G$. This is normalized via: $\eind(G) := \limsup_{m \to \infty} \frac{|\Aut(G)| \eind(G, m)}{(2m)^{\alpha^*(G)}},$ where $\Aut(G)$ is the automorphism group of $G$ and $\alpha^*(G)$ denotes the fractional independence number. This normalization aligns edge inducibility with the counting of $r$-cliques (Kruskal–Katona theorem) and links it to vertex inducibility (the maximum density of induced $G$ in large $n$-vertex hosts).

2. Hardness Reductions and Computational Equivalence

A major result connects the edge inducibility problem to vertex inducibility through a construction that “lifts” $G$ via adding pendant edges. For any graph $G$ on $n$ vertices, let $G'$ be obtained by attaching a pendant edge to each vertex. Then: $\frac{\ind(G)}{2^n\, n!} = \frac{\eind(G')}{|\Aut(G')|}.$ This equivalence implies that the computational complexity of edge inducibility (even for simple graphs) is at least as difficult as the vertex-based problem, carrying the known hardness and complexity properties of the latter into the edge inducibility setting.

3. Exact Edge-Inducibility for Small Graphs (≤4 vertices)

The inducibility has been determined for all graphs on at most 4 vertices. For these cases, explicit extremal hosts and densities have been constructed and proved:

• $\eind(K_{1,2}) = 1/4$ (star on 3 vertices)
• $\eind(K_{1,3}) = 1/8$ (star on 4 vertices)
• $\eind(C_4) = 1/2$ (4-cycle)
• Disjoint edges: $\eind$(pair of isolated edges) $= 1$
• For complete graphs ($K_2$, $K_3$, $K_4$): maximum attained by the complete host.

For non-clique graphs such as the path $P_4$, extremal constructions combine matchings with cliques to stitch together the desired induced copy density, resulting in $\eind(P_4) = 1/4$.

4. Edge-Inducibility for Larger Graphs and Special Structures

The analysis is extended to larger structures, notably the 5-cycle ($C_5$) and 6-vertex path ($P_6$):

• $C_5$:

$\eind(C_5) \leq \frac{1}{\sqrt{250}} \approx \frac{1}{15.8}.$

• $P_6$:

$\frac{5}{372} \leq \eind(P_6) \leq \frac{1}{36}.$

These bounds arise from entropy arguments and optimized constructions, with conjectured sharpness in the lower bounds.

5. Locally Directed Graphs: Structural Reductions and Generalizations

A novel contribution is the introduction of locally directed graphs (“local digraphs”). In these, each edge is assigned a local orientation (‘+’ or ‘–’) at each endpoint, rather than a global direction. When $G$ possesses a perfect matching that serves as its unique fractional perfect matching, a canonical local digraph $\ldg(G, M)$ is defined on the matching edges. The following relationship links edge inducibility with the acyclic vertex inducibility $\aind(\ldg(G, M))$: $\aind(\ldg(G, M)) \leq \frac{2^{|M|}|M|!}{|\Aut(G)|} \eind(G) \leq \ind(\ldg(G, M)).$ For instance, $P_4$ has unique matching $\{e_1, e_2\}$ and $\ldg(P_4, M)$ is an isolated edge, yielding $\eind(P_4) = 1/4$ as the vertex inducibility of an edge is 1.

6. Synthesis of Hardness and Structure Results

The reduction from vertex to edge inducibility via pendant-edge lift, combined with the locally directed graph framework, demonstrates that edge inducibility is not “easier” than vertex inducibility. For graphs with small size and sufficient symmetry (e.g. stars, cycles of length 4), full determination is feasible and explicit extremal hosts are available. For more complex structures (e.g. cycles of length 5 or paths of length 6), entropy-based inequalities and refined constructions yield nontrivial but still tractable bounds. The locally directed graph generalization enables further reductions and essentially captures the structure of perfect matching graphs in the inducibility context.

7. Connections, Broader Implications, and Future Directions

The equivalence between edge and vertex inducibility prompts renewed paper of induced density optimization in host graphs with fixed edge count, especially for larger and more complex patterns. The locally directed graph framework, in particular, opens the possibility of analyzing combinatorial parameters derived from fractional matchings, and can be extended to upper tail probability and extremal questions in random graphs and hypergraphs.

A notable inference is that an old conjecture of Erdős–Sós for 3-uniform hypergraphs (bounding edge count under bipartite link condition) is equivalent to $\ind(\mathrm{LDC}_3) = 1/4$ for the locally directed 3-cycle. Additionally, comparison of directed path inducibility in DAGs versus general digraphs reveals the acyclic value $\aind(DP_3) = 1/4$.

The results underscore the centrality of automorphism-based normalization and fractional combinatorial parameters in both edge and vertex inducibility. The sophisticated interplay between extremal graph theory, entropy methods, and local orientations suggests fertile ground for deeper structural results and algorithmic development, particularly in relation to upper tail counts in probabilistic combinatorics and the generalization of inducibility to other graph classes and higher-order structures.