On uniquely partitionable planar graphs

Let ~1,22 . . . . . ~,; n/>2 be any properties of graphs. A vertex (~L, ~2 . . . . . J~,,)-partition of a graph G is a partition (V1, l~,...,/7,,) of V(G) such that for each i = 1,2 . . . . . n the induced subgraph G[Vi] has the property ~i. A graph G is said to be uniquely (~1,~2 . . . . . ~,)-partitionable if G has unique vertex (2~1, ~2 , . . . , ~,)-partition. In the present paper we investigate the problem of the existence of uniquely (~1,~2 . . . . , ~n)-partitionable planar graphs for additive and hereditary properties ~1, ~2 , . . . , ~ , of graphs. Some constructions and open problems are presented for n = 2. (~) 1998 Elsevier Science B.V. All rights reserved