Testing for a Theta

Recently, a few new results have appeared, providing polynomial time algorithms 
for testing if a given graph contains certain induced subgraphs (such as
"pyramids", odd odd cycles and anticycles, and some others). However, some seemingly similar
problems (such as testing for the presence of an induced cycle passing through two given
vertices, or testing for "prisms") are known to be NP-complete. At the moment it is not
clear what causes this difference.

A "theta" is a graph consisting of three vertex disjoint induced paths
between two fixed vertices (the "ends"), such that there are no edges
between the interiors of different paths. In joint work with Paul Seymour we
were able to find a polynomial time algorithm to test if a graph contains a theta. In fact,
we prove a stronger result, that provides a necessary and sufficient condition for a graph
to contain a theta with a given end. We prove that a  graph G does not contain a
theta with a  given  end v, if and only if G has a certain structure; which can be tested
for in polynomial time.