The existence of continuations for different types of metrics

The problems of continuation of a partially defined metric and apartially defined ultrametric were considered in [12] and [13], respectively. Usingthe language of graph theory we generalize the criteria of existence of continuationobtained in these papers. For these purposes we use the concept of a trianglefunction introduced by M. Bessenyei and Z. Páles in [6], which gives a generalizationof the triangle inequality in metric spaces. The obtained result allowsus to get criteria of the existence of continuation for a wide class of semimetricsincluding not only metrics and ultrametrics, but also multiplicative metrics andsemimetrics with power triangle inequality. Moreover, the explicit formula for themaximal continuations is also obtained.