On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G − x which is point determining, is isomorphic to H . We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations. A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f(u)f(v) is an edge of H . For a fixed graph H , a full-H-colouring of G is a full homomorphism of G to H . A minimal H-obstruction is a graph G which does not admit a full Hcolouring, such that each proper induced subgraph of G admits a full H-colouring. We analyze minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k+ 1 vertices is a minimal H-obstruction if and only if it is a realization of H . We conclude that every minimal H-obstruction has at most k+1 vertices, and there are at most two minimal H-obstructions with k + 1 vertices. We also consider full homomorphisms to graphs H in which loops are allowed. If H has l loops and k vertices without loops, then every minimal H-obstruction has at most (k + 1)(l+ 1) vertices, and, when