The competitive diffusion game is a game-theoretic model of information spreading on a graph proposed by Alon et al. (2010). In the model, a player chooses an initial vertex of the graph, from which information by the player spreads through the edges connected with the initial vertex. If a vertex that is not yet influenced by any information receives information by a player, it is influenced by the information and it diffuses it to adjacent vertices. A vertex that simultaneously receives two or more types of information does not diffuse any type of information from then on. The objective of a player is to maximize the number of vertices influenced by the player’s information. In this paper, we investigate the existence of a pure Nash equilibrium of the two-player competitive diffusion game on chordal and its related graphs. We show that a pure Nash equilibrium always exists on block graphs, split graphs and interval graphs, all of which are well-known subclasses of chordal graphs. On the other hand, we show that there is an instance with no pure Nash equilibrium on (strongly) chordal graphs; the boundary of the existence of a pure Nash equilibrium is found.