On the geodetic number of median graphs (Q941326)

An interval \(I(u,v)\) in a graph is defined as the set of all vertices that lie on shortest paths between \(u\) and \(v\). A set \(S\) of vertices of a graph \(G\) is called geodetic if each vertex of \(G\) belongs to \(I(u,v)\) for some \(u,v\in S\). The minimal cardinality of a geodetic set in \(G\) is called the geodetic number of \(G\) and is denoted \(g(G)\). A graph \(G\) is called a median graph if for each triple of vertices \(u,v,w\) in \(G\) the intersection \(I(u,v) \cap I(u,w) \cap I(v,w)\) consists of precisely one vertex. The paper is devoted to geodetic sets of median graphs. The authors study how the function \(g(G)\) changes under the operation of peripheral expansion. Median graphs having geodetic sets of size two are characterized.