On a tree-partition problem

If T = ( V , E ) is a tree then – T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T 1 , T 2 , so that V ( T 1 ) ∩ V ( T 2 ) = { v } , V ( T 1 ) ∪ ( T 2 ) = V ( T ) , E ( T 1 ) ∩ E ( T 2 ) = ∅ , E ( T 1 ) ∪ E ( T 2 ) = E ( T ) , T [ V ( T 1 ) \ { v } ] ⊆ E ( T 1 ) and T [ V ( T 2 ) \ { v } ] ⊆ E ( T 2 ) . We call such a partition a T v − p a r t i t i o n of T. We study the following em: Given a graph G belonging to –T. Is it true that for any T v -partition T 1 , T 2 of T there exists a partition { V 1 , V 2 } of the vertices of G such that G [ V 1 ] ∈ − T 1 and G [ V 2 ] ∈ − T 2 ? This problem provides a natural generalization of Δ-partition problem studied by L. Lovász ([L. Lovász, On decomposition of graphs. Studia Sci. Math. Hungar. 1 (1966) 237–238]) and Path Partition Conjecture formulated by P. Mihók ([P. Mihók, Problem 4, in: M. Borowiecki, Z. Skupien (Eds.), Graphs, Hypergraphs and Matroids, Zielona Góra, 1985, p. 86]). We present some partial results and a contribution to the Path Kernel Conjecture that was formulated with connection to Path Partition Conjecture.