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Characterizing the interval function of a connected graph. (English) Zbl 0937.05036
Summary: As was shown in the book of H. M. Mulder [The interval function of a graph (Mathematical Centre Tracts 132, Mathematisch Centrum, Amsterdam) (1980; Zbl 0446.05039)], that 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 [Czech. Math. J. 44, No. 1, 173-178 (1994; Zbl 0808.05046)]. (Using the terminology of H.-J. Bandelt, M. van de Vel and E. Verheul [Math. Nachr. 163, 177-201 (1993; Zbl 0808.46011)] and H.-J. Bandelt and V. Chepoi [Discrete Math. 160, No. 1-3, 25-39 (1996; Zbl 0864.05049)], we may say that the author gave in 1994 a necessary and sufficient condition for a finite geometric interval space to be graphic.) In the present paper, the result given in 1994 is extended. The proof is based on new ideas.

05C12 Distance in graphs
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