Approximation algorithms for dominating set problem
The class of line graphs can be characterized by a finite family of forbidden induced subgraphs (Beineke).  A dominating set in a line graph G corresponds to an edge dominating set of the root graph of G (and vice versa), and the size of minimum edge dominating set can be approximated by factor of 2 in polynomial time.


http://en.wikipedia.org/wiki/Line_graph
http://en.wikipedia.org/wiki/Edge_dominating_set