Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. To be more formal, a graph G is a median graph if, for all μ, u, v ∈ V(G), it holds that |I(μ, u) ∩ I(μ, v) ∩ I(u, v)| = 1 where I(x, y) denotes the set of all vertices that lie on shortest paths connecting x and y.

In this paper we are interested in a natural generalization of median graphs, called k-median graphs. A graph G is a k-median graph, if there are k vertices μ₁, …, μ_k ∈ V(G) such that, for all u, v ∈ V(G), it holds that |I(μ_i, u) ∩ I(μ_i, v) ∩ I(u, v)| = 1, 1 ≤ i ≤ k. By definition, every median graph with n vertices is an n-median graph. We provide several characterizations of k-median graphs that, in turn, are used to provide many novel characterizations of median graphs.