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

Abstract

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 H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse 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 both k and ℓ are positive, there is at most one minimal H-obstruction with ( k + 1 ) ( ℓ + 1 ) vertices. In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.