On the Natural Imprint Function of a Graph

Abstract

In this paper, some characterizations of median and quasi-median graphs are extended to general isometric subgraphs of Cartesian products using the concept of an imprint function as introduced by Tardif. This extends the well known concepts of medians in median graphs as well as imprints in quasi-median graphs. We introduce absolute C-median graphs in analogy to absolute retracts, and derive a connection with the canonical isometric embedding of graphs into Cartesian products. Absolute C-median graphs strictly include classes of irreducible graphs and absolute (weak) retracts as well as many median-like classes, such as weakly median graphs, pre-median graphs, and weakly modular graphs. New characterizations of quasi-median graphs and of median graphs are obtained along the way. Finally, we propose a conjecture on the amalgamation procedure for absolute C-median graphs, and prove the fixed box theorem for this class modulo the conjecture.