We give a unified treatment of several fixed-point type theorems by using the concept of dismantlability, extended from ordered sets to arbitrary graphs. For a graph G and a vertex x of G we let NG(X) denote the set of neighbours of x in G. We say that x is a subdominant vertex of G if there is a vertex y of G, distinct from x, such that NG(x)∪{x} ⊆ NG(y)∪{y}. If G has n vertices we say that G is dismantlable if the vertices of G can be listed as x1, x2, ..., xi,..., xn such that, for all i = 1,2,..., n— 1, xi is a subdominant vertex of the graph Gi = G — {xj : j < i}
