The core of a graph

Abstract The core of a graph is its smallest subgraph which also is a homomorphic image. It turns out the core of a finite graph is unique (up to isomorphism) and is also its smallest retract. We investigate some homomorphism properties of cores and conclude that it is NP-complete to decide whether or not a graph is its own core. (A similar conclusion is reached about testing whether or not a graph is rigid, i.e., admits a non-identity homomorphism to itself.) We also give a polynomial-time verifiable condition for a graph of small independence number to be its own core.