On a generalization of median graphs: k-median graphs

04/13/2023

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by   Marc Hellmuth, et al.

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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 μ_1,…,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.