To delete vertices on a polygon mesh you must using Edit Mesh > Delete Edge/Vertex.
Many types of polygon vertices cannot be deleted using the key. For example, while you can delete border vertices on the exterior of a mesh using the key when the selected vertex shares only two edges, you cannot use the key to delete a vertex when it shares more than two edges.
When an interior vertex is deleted using Delete Edge/Vertex, any edges associated with the deleted vertices are also removed as a result. The polygon faces surrounding the deleted vertex are replaced with a single n-sided polygon created from the vertices surrounding the deleted vertex. Maya does not use quads and triangles in the affected area in order to reduce the number of polygons.
This action is equivalent to selecting all the edges surrounding the vertex and deleting them.
To delete vertices
To delete interior vertices