TLet G be a graph with a vertex set V. The graph G is path-proximinal if there is a semimetric d : V × V → [0, ∞[and disjoint proximinal subsets of the semimetric space (V, d) such that V = A ∪ B. The vertices u, v ∈ V are adjacent iff d(uv)⩽inf{d(xy):x∈Ay∈B}, and, for every p ∈ V , there is a path connecting A and B in G, and passing through p. It has been shown that a graph is path-proximinal if and only if all of its vertices are not isolated. It has also been shown that a graph is simultaneously proximinal and path-proximinal for an ultrametric if and only if the degree of each of its vertices is equal to 1.