Is the problem of finding the chromatic number of this modified interval graph NP-Complete?

Few days ago I was working on interval graphs to solve the known problem of resource allocation, as we know there is a greedy approach that solves this problem (chromatic number) in polynomial time and gives us the colors of each vertex i开发者_如何转开发n the interval graph (the problem of finding the chromatic number in general graphs is NP-Complete (3-satisfiability reduction by Karp)).

I was wondering: if had a graph that is not an interval graph but it is because it has one and only one chordless cycle of length > 3 (there is an edge that, when you remove it, the graph becomes an interval graph), does it makes the problem of finding the chromatic number on this kind of graph NP-Complete?

Kind of hand-wavey, if there's just a single edge which prevents the interval graph coloring algorithm from working, then remove it. Run the interval graph algorithm. If the two endpoints of the removed edge have different colors, you're done. Otherwise, Let C be the number of colors that the algorithm used. Try all (C choose 2) fixed colors for the two endpoints, and try the interval graph algorithm again. If it succeeds with C colors, you're done. Otherwise, you'll need C+1 colors so just pick one of the endpoints and give it a unique color.

I'm assuming you can find the removable edge in poly time.