Genus of a graph and its strong preservers.

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k−1. A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and that maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus not k to graphs of genus not k. We show that such linear operators are necessarily vertex permutations. [ABSTRACT FROM AUTHOR]