# The induced paths in a connected graph and a ternary relation determined by them

Mathematica Bohemica (2002)

• Volume: 127, Issue: 3, page 397-408
• ISSN: 0862-7959

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## Abstract

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By a ternary structure we mean an ordered pair $\left({X}_{0},{T}_{0}\right)$, 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 $\left({X}_{0},{T}_{0}\right)$ 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 $\left\{x\in {X}_{0}\phantom{\rule{0.277778em}{0ex}}{T}_{0}\left(u,x,v\right)\right\}\cup \left\{x\in {X}_{0}\phantom{\rule{0.277778em}{0ex}}{T}_{0}\left(v,x,u\right)\right\}=\left\{u,v\right\}.$ A ternary structure $\left({X}_{0},{T}_{0}\right)$ 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 $u,x,y\in {X}_{0}$: ${T}_{0}\left(x,u,y\right)$ if and only if $u$ belongs to an induced $x-y$ path in $G$. It is clear that if a ternary structure $\left({X}_{0},{T}_{0}\right)$ is the B-structure of a connected graph $G$, then $G$ is the underlying graph of $\left({X}_{0},{T}_{0}\right)$. We will prove that there exists no sentence $\sigma$ of the first-order logic such that a ternary structure $\left({X}_{0},{T}_{0}\right)$ with a connected underlying graph $G$ is the B-structure of $G$ if and only if $\left({X}_{0},{T}_{0}\right)$ satisfies $\sigma$.

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