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Discussiones Mathematicae Graph Theory  17(1) (1997)   95-102
DOI: https://doi.org/10.7151/dmgt.1042

PARTITIONS OF SOME PLANAR GRAPHS INTO TWO LINEAR FORESTS

Piotr Borowiecki
and
Mariusz Hałuszczak

Institute of Mathematics
Technical University of Zielona Góra
Podgórna 50, 65-246 Zielona Góra, Poland

Abstract

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.

1991 Mathematics Subject Classification: 05C15, 05C70.

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