On uniquely partitionable planar graphs Original Research Article

Let P1, P2, …, Pn; n ⩾ 2 be any properties of graphs. A vertex (P1, P2, …, Pn)-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 Pi. A graph G is said to be uniquely (P1, P2, …, Pn)-partitionable if G has unique vertex (P1, P2, …, Pn)-partition. In the present paper we investigate the problem of the existence of uniquely (P1, P2, …, Pn)-partitionable planar graphs for additive and hereditary properties P1, P2, …, Pn of graphs. Some constructions and open problems are presented for n = 2.