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The induced path function, monotonicity and betweenness. (English) Zbl 1225.05146
Summary: The geodesic interval function $$I$$ of a connected graph allows an axiomatic characterization involving axioms on the function only, without any reference to distance, as was shown by L. Nebeský [Czech. Math. J. 44, No.1, 173-178 (1994; Zbl 0808.05046)]. Surprisingly, L. Nebeský [Math. Bohem. 127, No. 3, 397–408 (2002; Zbl 1003.05063)] showed that, if no further restrictions are imposed, the induced path function $$J$$ of a connected graph $$G$$ does not allow such an axiomatic characterization. Here $$J(u,v)$$ consists of the set of vertices lying on the induced paths between $$u$$ and $$v$$. This function is a special instance of a transit function.
In this paper we address the question what kind of restrictions could be imposed to obtain axiomatic characterizations of $$J$$. The function $$J$$ satisfies betweenness if $$w \in J(u,v)$$, with $$w \neq u$$, implies $$u \notin J(w,v)$$ and $$x \in J(u,v)$$ implies $$J(u,x) \subseteq J(u,v)$$. It is monotone if $$x,y \in J(u,v)$$ implies $$J(x,y) \subseteq J(u,v)$$. In the case where we restrict ourselves to functions $$J$$ that satisfy betweenness, or monotonicity, we are able to provide such axiomatic characterizations of $$J$$ by transit axioms only. The graphs involved can all be characterized by forbidden subgraphs.

##### MSC:
 05C38 Paths and cycles 03C13 Model theory of finite structures
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##### References:
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