Role colouring graphs in hereditary classes

A locally surjective homomorphism from a graph G to H is called an -role colouring of . Deciding the existence such with | = k known be NP -hard even under substantial restrictions on input We study family hereditary classes for which problem can solved efficiently. Our main result first boundary class this problem; that is, minimal obstruction solving efficiently in finitely defined class. also give related -coupon problem, complete each vertex incident loop, when prime power. As additional results, we show 2-role 2 K -free graphs and 2-coupon cographs done linear time.