Uniquely partitionable planar graphs with respect to properties having a forbidden tree

Abstract

Let P 1 , P 2 be graph properties. A vertex (P 1 , P 2 )-partition of a graph G is a partition {V 1 , V 2 } of V (G) such that for i = 1, 2 the induced subgraph G[V i ] has the property P i . A property R = P 1 •P 2 is defined to be the set of all graphs having a vertex (P 1 , P 2 )-partition. A graph G ∈ P 1 •P 2 is said to be uniquely (P 1 , P 2 )-partitionable if G has exactly one vertex (P 1 , P 2 )-partition. In this note, we show the existence of uniquely partitionable planar graphs with respect to hereditary additive properties having a forbidden tree.

Remarks on the existence of uniquely partitionable planar graphs

“…. , P n )-partitionable planar graphs has been investigated in [9] and [8]. In this section we summarize the known results: Proposition 1 [9].…”