Topic
K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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01 Jan 2014
TL;DR: In this article, the authors considered the problem of partitioning a set of points in R into (d+1) parts of strictly smaller diameter, and showed that the positive answer to Borsuk's conjecture is true in dimensions up to 3.
Abstract: Note that we assume of the sphere being embedded in R, and the unit distance included from the ambient space. Diameter graphs arise naturally in the context of Borsuk’s problem. In 1933 Borsuk [3] asked whether any set of diameter 1 in R can be partitioned into (d+1) parts of strictly smaller diameter. The positive answer to this question is called Borsuk’s conjecture. This was shown to be true in dimensions up to 3. In 1993 Kahn and Kalai [6] constructed a finite set of points in dimensions 1325 that does not admit a partition into 1326 parts of smaller diameter. The minimal dimension in which the counterexample is known is 64 (see [2], [5]). We focus on one conjecture, posed by Morić and Pach [12].
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TL;DR: It is shown how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph and new cliques in the complement graph can be detected by a condition on labels for any labeling structure.
Abstract: Algorithm MLS (Maximal Label Search) is a graph search algorithm which generalizes algorithms MCS, LexBFS, LexDFS and MNS On a chordal graph, MLS computes a peo (perfect elimination ordering) of the graph We show how algorithm MLS can be modified to compute a pmo (perfect moplex ordering) as well as a clique tree and the minimal separators of a chordal graph We give a necessary and sufficient condition on the labeling structure for the beginning of a new clique in the clique tree to be detected by a condition on labels MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure A linear time algorithm computing a pmo and the generators of the maximal cliques and minimal separators wrt this pmo of the complement graph is provided On a non-chordal graph, algorithm MLSM is used to compute an atom tree of the clique minimal separator decomposition of any graph
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TL;DR: This work presents a 3-approximation for the unweighted case and a 4-app approximation for the weighted case with nonnegative weights for line graphs, a ?
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TL;DR: This paper gives a new algorithm that first computes and then repeatedly augments a spanning chordal subgraph and proves that the algorithm terminates with a maximal chordalSubgraph, and demonstrates that this algorithm is more amenable to parallelization and that the parallel version also ends with aMaximum chordal Subgraph.
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TL;DR: In this paper, it was shown that for a connected graph G, if for any cut-set, then G has a k-tree, then it is a spanning tree with maximum degree at most k.
Abstract: A k-tree of a connected graph G is a spanning tree with maximum degree
at most k. The rupture degree for a
connected graph G is defined by , where and ,
respectively, denote the order of the largest component and number of
components in . In this
paper, we show that for a connected graph G,
if for any cut-set , then G has a k-tree.