  Consider the following problem. A seller has infinite copies of $n$ products
represented by nodes in a graph. There are $m$ consumers, each has a budget and
wants to buy two products. Consumers are represented by weighted edges. Given
the prices of products, each consumer will buy both products she wants, at the
given price, if she can afford to. Our objective is to help the seller price
the products to maximize her profit.
  This problem is called {\em graph vertex pricing} ({\sf GVP}) problem and has
resisted several recent attempts despite its current simple solution. This
motivates the study of this problem on special classes of graphs. In this
paper, we study this problem on a large class of graphs such as graphs with
bounded treewidth, bounded genus and $k$-partite graphs.
  We show that there exists an {\sf FPTAS} for {\sf GVP} on graphs with bounded
treewidth. This result is also extended to an {\sf FPTAS} for the more general
{\em single-minded pricing} problem. On bounded genus graphs we present a {\sf
PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs.
  We study the Sherali-Adams hierarchy applied to a natural Integer Program
formulation that $(1+\epsilon)$-approximates the optimal solution of {\sf GVP}.
Sherali-Adams hierarchy has gained much interest recently as a possible
approach to develop new approximation algorithms. We show that, when the input
graph has bounded treewidth or bounded genus, applying a constant number of
rounds of Sherali-Adams hierarchy makes the integrality gap of this natural
{\sf LP} arbitrarily small, thus giving a $(1+\epsilon)$-approximate solution
to the original {\sf GVP} instance.
  On $k$-partite graphs, we present a constant-factor approximation algorithm.
We further improve the approximation factors for paths, cycles and graphs with
degree at most three.
