Topic
K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
Papers published on a yearly basis
Papers
More filters
01 Jan 2006
TL;DR: This work considers the problem of generating all maximal cliques in an unit disk graph, and considers several characteristic shapes determined by that edge, and proves that all cliques having this as the longest edge are included in one of the sets of vertices contained in these shapes.
Abstract: We consider the problem of generating all maximal cliques in an unit disk graph. General algorithms to find all maximal cliques are exponential, so we rely on a polynomial approximation. Our algorithm makes use of certain key geographic structures of these graphs. For each edge, we limit the set of vertices that may form cliques with this as the longest edge. We then consider several characteristic shapes determined by that edge, and prove that all cliques having this as the longest edge, are included in one of the sets of vertices contained in these shapes. Our algorithm works in O(m¢ 2 ) time and generates O(m¢) cliques, where m is the number of edges in the graph and ¢ is its maximum degree. We also provide a modified version of the algorithm which improves the performance in many cases, albeit without affecting the worst case running time.
36 citations
TL;DR: An NP-complete communication problem on tree networks, the so-called MULTICUT in TREES, is studied, and a simple fixed-parameter algorithm is provided showing fixed- Parameter tractability with respect to parameter k.
Abstract: We study an NP-complete (and MaxSNP-hard) communication problem on tree networks, the so-called MULTICUT IN TREES: given an undirected tree and some pairs of nodes of the tree, find out whether there is a set of at most k tree edges whose removal separates all given pairs of nodes. MULTICUT has been intensively studied for trees as well as for general graphs mainly from the viewpoint of polynomial time approximation algorithms. By way of contrast, we provide a simple fixed-parameter algorithm for MULTICUT IN TREES showing fixed-parameter tractability with respect to parameter k. Moreover, based on some polynomial time data reduction rules, which appear to be of particular interest from an applied point of view, we show a problem kernel for MULTICUT IN TREES by an intricate mathematical analysis. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(3), 124–135 2005
35 citations
TL;DR: This paper presents a novel approach for partial clique enumeration, which is based on a continuous formulation of the clique problem developed in the 1960s by Motzkin and Straus, and is able to avoid extracting the same clique multiple times.
34 citations
TL;DR: The findings indicate that there are considerable variations between the centrality values of large cliques within the same networks, and the most central cliques in graphs are not necessarily the largest ones.
Abstract: We consider the problem of finding the most and the least “influential” or “influenceable” cliques in graphs based on three classical centrality measures: degree, closeness and betweenness. In addition to standard clique betweenness, which is defined as the proportion of shortest paths between any two graph nodes that pass through the clique, we also consider its optimistic and pessimistic versions, where outside nodes may favor or try to avoid shortest paths passing through the clique, respectively. We discuss the computational complexity issues for these problems and develop linear 0–1 programming formulations for each centrality metric. Finally, we demonstrate the performance of the developed formulations on real-life and synthetic networks, and provide some interesting insights based on the obtained results. In particular, our findings indicate that there are considerable variations between the centrality values of large cliques within the same networks. Moreover, the most central cliques in graphs are not necessarily the largest ones.
34 citations
TL;DR: A Lagrangian based exact solution algorithm for the vehicle routing problem (VRP), defined on an undirected graph, is introduced in this paper and lower bounds are obtained by allowing exponentially many inequalities as candidates tolagrangian dualization.
33 citations