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graphs; distance; interval function
As was shown in the book of Mulder [4], the interval function is an important tool for studying metric properties of connected graphs. An axiomatic characterization of the interval function of a connected graph was given by the present author in [5]. (Using the terminology of Bandelt, van de Vel and Verheul [1] and Bandelt and Chepoi [2], we may say that [5] gave a necessary and sufficient condition for a finite geometric interval space to be graphic). In the present paper, the result given in [5] is extended. The proof is based on new ideas.
[1] H.-J. Bandelt M. van de Vel E.Verheul: Modular interval spaces. Math. Nachr. 163 (1993), 177-201. DOI 10.1002/mana.19931630117 | MR 1235066
[2] H.-J. Bandelt V. Chepoi: A Holly theorem in weakly modular space. Discrete Math. 160 (1996), 25-39. DOI 10.1016/0012-365X(95)00217-K | MR 1417558
[3] G. Chartrand L. Lesniak: Graphs & Digraphs. (third edition). Chapman & Hall, London, 1996. MR 1408678
[4] H. M. Mulder: The Interval Function of a Graph. Mathematisch Centrum, Amsterdam, 1980. MR 0605838 | Zbl 0446.05039
[5] L. Nebeský: A characterization of the interval function of a connected graph. Czechoslovak Math. J. 44 (119) (1994), 173-178. MR 1257943
[6] L. Nebeský: Geodesics and steps in a connected graph. Czechoslovak Math. J. 47 (122) (1997), 149-161. DOI 10.1023/A:1022404624515 | MR 1435613
[7] E. R. Verheul: Multimedians in metric and normed spaces. CWI TRACT 91, Amsterdam, 1993. MR 1244813 | Zbl 0790.46008
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