Directed by Royal Court associate Jeremy Herrin, the production features a set that recreates the iconic Commons benches and an ensemble cast that includes Charles Edwards (pictured below) and Philip Glenister.
Precise dates and casting for the Olivier run are still to be announced.
According to press material: "This House strips politics down to the practical realities of those behind the scenes; the whips who roll up their sleeves and on occasion bend the rules to shepherd and coerce a diverse chorus of MPs within the Mother of all Parliaments."
Graham won the Pearson Playwriting Bursary in 2006 and went on to win the Catherine Johnson Award for Best Play of 2007 for Eden's Empire. His other plays also include Tory Boyz for the National Youth Theatre and The Whisky Taster at the Bush.
This House, which continues its sold out run in the Cottesloe until 1 December, is designed by Rae Smith with lighting design by Paule Constable, music by Stephen Warbeck, choreography by Scott Ambler and sound by Ian Dickinson.