We say that a binary operation $*$ is associated with a (finite undirected) graph $G$ (without loops and multiple edges) if $*$ is defined on $V\left(G\right)$ and $uv\in E\left(G\right)$ if and only if $u\ne v$, $u*v=v$ and $v*u=u$ for any $u$, $v\in V\left(G\right)$. In the paper it is proved that a connected graph $G$ is geodetic if and only if there exists a binary operation associated with $G$ which fulfils a certain set of four axioms. (This characterization is obtained as an immediate consequence of a stronger result proved in the paper).

By a ternary system we mean an ordered pair $(W,R)$, where $W$ is a finite nonempty set and $R\subseteq W\times W\times W$. By a signpost system we mean a ternary system $(W,R)$ satisfying the following conditions for all $x,y,z\in W$: if $(x,y,z)\in R$, then $(y,x,x)\in R$ and $(y,x,z)\notin R$; if $x\ne y$, then there exists $t\in W$ such that $(x,t,y)\in R$. In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G,T)$, where $G$ is a connected graph and $T$ is a spanning tree of $G$. If $(G,T)$ is a ct-pair, then by...

By a chordal graph is meant a graph with no induced cycle of length $\ge 4$. By a ternary system is meant an ordered pair $(W,T)$, where $W$ is a finite nonempty set, and $T\subseteq W\times W\times W$. Ternary systems satisfying certain axioms (A1)–(A5) are studied in this paper; note that these axioms can be formulated in a language of the first-order logic. For every finite nonempty set $W$, a bijective mapping from the set of all connected chordal graphs $G$ with $V\left(G\right)=W$ onto the set of all ternary systems $(W,T)$ satisfying the axioms (A1)–(A5)...