Geodesic
A path(ological) partition problem
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 113-125

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Let τ(G) denote the number of vertices in a longest path of the graph G and let k₁ and k₂ be positive integers such that τ(G) = k₁ + k₂. The question at hand is whether the vertex set V(G) can be partitioned into two subsets V₁ and V₂ such that τ(G[V₁] ) ≤ k₁ and τ(G[V₂] ) ≤ k₂. We show that several classes of graphs have this partition property.
Keywords: vertex partition, τ-partitionable, decomposable graph 
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Broere, Izak; Dorfling, Michael; Dunbar, Jean; Frick, Marietjie. A path(ological) partition problem. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 1, pp. 113-125. http://geodesic.mathdoc.fr/item/DMGT_1998_18_1_a9/