We consider even order graphs in which no two points have more than one join and no point is joined to itself. In such a graph G, of order 2n, [A, B] denotes an "equipartition of G" if A and B are subgraphs of G of order n whose vertex sets are disjoint. When A is isomorphic to B the equipartition is a "bisection of G." We establish that for every integer n> 1 there exists a nontrivial graph of order 2n which "has all bisections," that is, a graph such that every equipartition is a bisection. (In this connection, a graph is trivial if it has all joins or no joins.) The class of graphs having all bisections is completely characterized.