Geodesic
Partitions of some planar graphs into two linear forests
Discussiones Mathematicae. Graph Theory, Tome 17 (1997) no. 1, pp. 95-102

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A linear forest is a forest in which every component is a path. It is known that the set of vertices V(G) of any outerplanar graph G can be partitioned into two disjoint subsets V₁,V₂ such that induced subgraphs 〈V₁〉 and 〈V₂〉 are linear forests (we say G has an (LF, LF)-partition). In this paper, we present an extension of the above result to the class of planar graphs with a given number of internal vertices (i.e., vertices that do not belong to the external face at a certain fixed embedding of the graph G in the plane). We prove that there exists an (LF, LF)-partition for any plane graph G when certain conditions on the degree of the internal vertices and their neighbourhoods are satisfied.
Keywords: linear forest, bipartition, planar graphs 
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Borowiecki, Piotr; Hałuszczak, Mariusz. Partitions of some planar graphs into two linear forests. Discussiones Mathematicae. Graph Theory, Tome 17 (1997) no. 1, pp. 95-102. http://geodesic.mathdoc.fr/item/DMGT_1997_17_1_a5/