On the Natural Imprint Function of a Graph

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 medianlike 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 amalgamation procedure for absolute C-median graphs, and prove the fixed box theorem for this class modulo the conjecture.