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connected graph; induced path; ternary relation; finite structure
By a ternary structure we mean an ordered pair $(X_0, T_0)$, where $X_0$ is a finite nonempty set and $T_0$ is a ternary relation on $X_0$. By the underlying graph of a ternary structure $(X_0, T_0)$ we mean the (undirected) graph $G$ with the properties that $X_0$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if \[\lbrace x \in X_0\; T_0(u, x, v)\rbrace \cup \lbrace x \in X_0\; T_0(v, x, u)\rbrace = \lbrace u, v\rbrace .\] A ternary structure $(X_0, T_0)$ is said to be the B-structure of a connected graph $G$ if $X_0$ is the vertex set of $G$ and the following statement holds for all $u, x, y \in X_0$: $T_0(x, u, y)$ if and only if $u$ belongs to an induced $x-y$ path in $G$. It is clear that if a ternary structure $(X_0, T_0)$ is the B-structure of a connected graph $G$, then $G$ is the underlying graph of $(X_0, T_0)$. We will prove that there exists no sentence $\sigma $ of the first-order logic such that a ternary structure $(X_0, T_0)$ with a connected underlying graph $G$ is the B-structure of $G$ if and only if $(X_0, T_0)$ satisfies $\sigma $.
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