Extremal values for identification, domination and maximum cliques in twin-free graphs

Consider a connected undirected graph G = (V, E) and an integer r ≥ 1; for any vertex v G V, let B_r(v) denote the ball of radius r centered at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ε V, the sets B_r(v) are different, then we say that G is r-twin-free. Studies have been made, e.g., on the number of edges or the minimum degree in one-twin-free graphs. We extend these investigations and in particular we determine the exact size of the largest clique in a connected r-twin-free graph.