We consider a competitive facility location problem on a network, in which consumers are located on the vertices and wish to connect to the nearest facility. Knowing this, competitive players locate their facilities on vertices that capture the largest possible market share. In 1991, Eiselt and Laporte established the first relation between Nash equilibria of a facility location game in a duopoly and the solutions to the 1-median problem. They showed that an equilibrium always exists in a tree because a location profile is at equilibrium if and only if both players select a 1-median of that tree . In this work, we further explore the relations between the solutions to the 1-median problem and the equilibrium profiles. We show that if an equilibrium in a cycle exists, both players must choose a solution to the 1-median problem. We also obtain the same property for some other classes of graphs such as quasi-median graphs, median graphs, Helly graphs, and strongly-chordal graphs. Finally, we prove the converse for the latter class, establishing that, as for trees, any median of a strongly-chordal graph is a winning strategy that leads to an equilibrium.