A graph G is said to be uniquely H-saturated if it contains no H, but adding any edge to G creates exactly one copy of H

Definitions and mathematical statement

Fix an integer $r\ge3$. A finite simple graph $G$ is uniquely $K_r$-saturated if it contains no copy of the complete graph $K_r$, but adding any missing edge creates exactly one copy of $K_r$. Copies are counted by their vertex sets, and every newly created copy necessarily uses the added edge. A dominating vertex is one adjacent to every other vertex.

Call a uniquely $K_r$-saturated graph sporadic here if it has no dominating vertex. Cooper's finiteness question is whether, for every fixed $r\ge3$, there are only finitely many isomorphism classes of sporadic uniquely $K_r$-saturated graphs. Equivalently, does there exist a bound $N(r)$ such that \[ |V(G)|\le N(r) \] for every such graph without a dominating vertex?

The bound may depend on $r$ but not on the graph. Requiring exactly one newly formed clique is stronger than ordinary saturation, which merely requires at least one. The absence of a dominating vertex removes familiar infinite families and is part of the question, not a cosmetic normalization. The restriction $r\ge3$ also matters: for $r=2$, edgeless graphs of arbitrary order already make the analogous finiteness assertion false. Counting labeled graphs rather than isomorphism types does not change finiteness once their orders are bounded.

Short English statement

For each fixed r at least three, are there finitely many uniquely Kr-saturated graphs without a dominating vertex?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3100025 (AMR-030-0025), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Cooper - Combinatorial Problems I Like (2020 snapshot), Graphs, Hypergraphs, Set Systems, Designs, source-order bullet 25. Original source indexed by AMR-030-0025. https://people.math.sc.edu/cooper/combprob.html.
• S3 Electronic Journal of Combinatorics 25(4), paper35, uniquely saturated graphs and Cooper’s finiteness question. https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p35/pdf/.