On uniquely partitionable planar graphs

Abstract Let P 1, P 2, …, P n; n ⩾ 2 be any properties of graphs. A vertex ( P 1, P 2, …, P n)-partition of a graph G is a partition (V1, V2, …, Vn) of V(G) such that for each i = 1, 2, …, n the induced subgraph G[Vi] has the property P i. A graph G is said to be uniquely ( P 1, P 2, …, P n)-partitionable if G has unique vertex ( P 1, P 2, …, P n)-partition. In the present paper we investigate the problem of the existence of uniquely ( P 1, P 2, …, P n)-partitionable planar graphs for additive and hereditary properties P 1, P 2, …, P n of graphs. Some constructions and open problems are presented for n = 2.