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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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