Characterization of Graphical Apiculate and Generalized Median Algebras

Definitions and mathematical statement

A ternary algebra is a set $A$ equipped with an operation $\alpha\colon A^3\to A$. A subset $B\subseteq A$ is closed under $\alpha$ if \[ \forall u,v,w\in B,\qquad \alpha(u,v,w)\in B. \] A subalgebra carries the restricted operation $\alpha|_{B^3}$. The supplied record does not specify whether empty carriers or empty subalgebras are allowed; the quantification over subalgebras below is subject to that unresolved convention.

The record calls a ternary algebra apiculate if it is realized as the apex algebra of an apiculate graph. It calls an algebra $\mathcal A$ graphical if $\mathcal A$ itself and every subalgebra of $\mathcal A$ are apiculate algebras. Thus, writing $\operatorname{Apic}(\mathcal A)$ for the supplied representability condition, \[ \operatorname{Graphical}(\mathcal A) \iff \operatorname{Apic}(\mathcal A) \ \land\ \left(\forall\mathcal B\text{ a subalgebra of }\mathcal A,\ \operatorname{Apic}(\mathcal B)\right). \] The original excerpt says that every connected graph $G=(V,E)$ admits an apex operation $V^3\to V$, but its description of that operation is truncated. Neither an apex operation nor the defining conditions for an apiculate graph, a generalized median graph, or a generalized median algebra are supplied completely. Consequently these terms remain source-dependent placeholders here.

Requested characterization, attributed in the supplied heading to Chepoi & Bandelt (Problem 14). Characterize the apiculate graphs whose apex algebras are graphical. Likewise, characterize the generalized median graphs whose generalized median algebras are graphical. Symbolically, if $\operatorname{Apex}(G)$ and $\operatorname{GM}(G)$ denote the associated algebras intended by the source, the requested classes are \[ \begin{aligned} &\{G: G\text{ is an apiculate graph and } \operatorname{Graphical}(\operatorname{Apex}(G))\},\\ &\{G: G\text{ is a generalized median graph and } \operatorname{Graphical}(\operatorname{GM}(G))\}. \end{aligned} \] These displays express the supplied targets, not complete intrinsic definitions. The record does not determine whether graphicality in the second question retains the apiculate representability condition above or instead means representability, for the algebra and all its subalgebras, by generalized median graphs. It also does not specify any quantifier over choices of associated operations if those operations are not unique. These issues require resolution from the complete mathematical source before a fully self-contained statement can be accepted.

Short English statement

The supplied question asks which apiculate graphs have apex algebras that, together with every subalgebra, are realizable as apex algebras of apiculate graphs. It asks an analogous question for generalized median graphs. Essential graph and operation definitions are missing, and the meaning of graphicality in the second question is unresolved. The supplied open label is a curation claim, not an independently verified mathematical conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006626$, exact source problem code catalogue entry, title: Characterization of Graphical Apiculate and Generalized Median Algebras. The supplied original heading identifies Problem 14 and attributes it to Chepoi & Bandelt; the structured proposer field is empty.
• Median Geometry and Applications (2026), Oberwolfach Reports, pp. 487--540, DOI 10.4171/owr/2026/8. Supplied source link: https://doi.org/10.4171/owr/2026/8.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/53603.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the record contains only the setup of Problem 14, without a complete claim to verify, while the associated characterization questions remain open. This is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired for self-contained publication. These are supplied verification claims; this draft does not independently confirm them.