k-Path Partitions in Trees (original) (raw)
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Lecture Notes in Computer Science, 2007
In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint 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 , which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree 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 NP-hardness of the k-tree 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 .
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Vertex Covering by Paths on Trees with applications in machine translation is the task to cover all vertices of a tree T = (V, E) by choosing a minimum-weight subset of given paths in the tree. The problem is NP-hard and has recently been solved by an exact algorithm running in O(4 C · |V | 2 ) time, where C denotes the maximum number of paths covering a tree vertex. We improve this running time to O(2 C · C · |V |). On the route to this, we introduce the problem Tree-like Weighted Hitting Set which might be of independent interest. In addition, for the unweighted case of Vertex Covering by Paths on Trees, we present an exact algorithm using a search tree of size O(2 k · k!), where k denotes the number of chosen covering paths. Finally, we briefly discuss the existence of a size-O(k 2 ) problem kernel.
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Graph partitioning is a widely studied problem in the literature with several applications in real life contexts. In this paper we study the problem of partitioning a graph, with weights at its vertices, into p connected components. For each component of the partition we measure the difference between the maximum and the minimum weight of a vertex in the component. We consider two objective functions to minimize, one measuring the maximum of such differences among all the components in the partition, and the other measuring the sum of the differences between the maximum and the minimum weight of a vertex in each component. We focus our analysis on tree graphs and provide polynomial time algorithms for solving these optimization problems on such graphs. In particular, we present an O(n 2 log n) time algorithm for the min-max version of the problem on general trees and several, more efficient polynomial algorithms for some trees with a special structure, such as spiders and caterpillars. Finally, we present NP-hardness and approximation results on general graphs for both the objective functions.
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On covering vertices of a graph by trees
Discrete Mathematics, 2008
The purpose of this paper is to initiate study of the following problem: Let G be a graph, and k 1. Determine the minimum number s of trees T 1 ,. .. , T s , (T i) k, i = 1,. .. , s, covering all vertices of G. We conjecture: Let G be a connected graph, and k 2. Then the vertices of G can be covered by s n− (k−1)+1 edge-disjoint trees of maximum degree k. As a support for the conjecture we prove the statement for some values of and k.
2010
A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ψ k (G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining ψ k (G) is NP-hard for each k ≥ 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of ψ k (G) and provide several estimations and exact values of ψ k (G). We also prove that ψ 3 (G) ≤ (2n + m)/6, for every graph G with n vertices and m edges.