  The problem of finding an optimal vertex cover in a graph is a classic
NP-complete problem, and is a special case of the hitting set question. On the
other hand, the hitting set problem, when asked in the context of induced
geometric objects, often turns out to be exactly the vertex cover problem on
restricted classes of graphs. In this work we explore a particular instance of
such a phenomenon. We consider the problem of hitting all axis-parallel slabs
induced by a point set P, and show that it is equivalent to the problem of
finding a vertex cover on a graph whose edge set is the union of two
Hamiltonian Paths. We show the latter problem to be NP-complete, and we also
give an algorithm to find a vertex cover of size at most k, on graphs of
maximum degree four, whose running time is 1.2637^k n^O(1).
