On uniquely partitionable planar graphs (Q1584417)

Let \({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n\); \(n\geq 2\) be any properties of graphs. A vertex \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition of a graph \(G\) is a partition \((V_1,V_2,\dots, V_n)\) of \(V(G)\) such that for each \(i= 1,2,\dots, n\) the induced subgraph \(G[V_i]\) has the property \({\mathcal P}_i\). A graph \(G\) is said to be uniquely \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partitionable if \(G\) has unique vertex \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partition. In the present paper we investigate the problem of the existence of uniquely \(({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n)\)-partitionable planar graphs for additive and hereditary properties \({\mathcal P}_1,{\mathcal P}_2,\dots,{\mathcal P}_n\) of graphs. Some constructions and open problems are presented for \(n= 2\).