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Partitions of graphs into trees
 IN PROCEEDINGS OF GRAPH DRAWING’06 (KARLSRUHE), VOLUME 4372 OF LNCS
, 2007
"... In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) ..."
Abstract

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In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3tree partition of the inner edges. Maximal planar bipartite graphs have a 2tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NPhardness of the ktree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].