Edge integrity of nearest neighbor graphs and separator theorems

We give upper bounds for the edge integrity and the vertex integrity of $k$-type nearest neighbor graphs in ${\mathbb{R}}^d$. For $n$ the order of a graph, examples are constructed to show that the bounds are best possible in the parameters $n$ and $k$. The results follow from decompositions that are shown for certain graphs that possess small edge or vertex separators. A vertex separator theorem for nearest neighbor graphs is known, and an edge separator theorem is shown. We also apply the decompositions to obtain upper bounds for the edge integrity of planar graphs, graphs of fixed genus, and trees.