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 ( V 1, V 2, …, V n ) of V( G) such that for each i = 1, 2, …, n the induced subgraph G[ V i ] 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.
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