Abstract
We find a set of necessary and sufficient conditions under which the weight \({w: E \rightarrow \mathbb{R}^{+}}\) on the graph G = (V, E) can be extended to a pseudometric \({d : V \times V \rightarrow \mathbb{R}^{+}}\). We describe the structure of graphs G for which the set \({\mathfrak{M}_{w}}\) of all such extensions contains a metric whenever w is strictly positive. Ordering \({\mathfrak{M}_{w}}\) by the pointwise order, we have found that the posets \(({\mathfrak{M}_{w}, \leqslant)}\) contain the least elements ρ 0,w if and only if G is a complete k-partite graph with \({k \, \geqslant \, 2}\). In this case the symmetric functions \({f : V \times V \rightarrow \mathbb{R}^{+}}\), lying between ρ 0,w and the shortest-path pseudometric, belong to \({\mathfrak{M}_{w}}\) for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.
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Dovgoshey, O., Martio, O. & Vuorinen, M. Metrization of Weighted Graphs. Ann. Comb. 17, 455–476 (2013). https://doi.org/10.1007/s00026-013-0192-7
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DOI: https://doi.org/10.1007/s00026-013-0192-7