2,696 research outputs found
Interval Routing Schemes for Circular-Arc Graphs
Interval routing is a space efficient method to realize a distributed routing
function. In this paper we show that every circular-arc graph allows a shortest
path strict 2-interval routing scheme, i.e., by introducing a global order on
the vertices and assigning at most two (strict) intervals in this order to the
ends of every edge allows to depict a routing function that implies exclusively
shortest paths. Since circular-arc graphs do not allow shortest path 1-interval
routing schemes in general, the result implies that the class of circular-arc
graphs has strict compactness 2, which was a hitherto open question.
Additionally, we show that the constructed 2-interval routing scheme is a
1-interval routing scheme with at most one additional interval assigned at each
vertex and we an outline algorithm to calculate the routing scheme for
circular-arc graphs in O(n^2) time, where n is the number of vertices.Comment: 17 pages, to appear in "International Journal of Foundations of
Computer Science
On the space requirement of interval routing
Interval routing is a space-efficient method for point-to-point networks. It is based on labeling the edges of a network with intervals of vertex numbers (called interval labels). An M-label scheme allows up to M labels to be attached on an edge. For arbitrary graphs of size n, n the number of vertices, the problem is to determine the minimum M necessary for achieving optimality in the length of the longest routing path. The longest routing path resulted from a labeling is an important indicator of the performance of any algorithm that runs on the network. We prove that there exists a graph with D = Ω(n1/3) such that if M †n/18D - O(ân/D), the longest path is no shorter than D + Î(D/âM). As a result, for any M-label IRS, if the longest path is to be shorter than D + Î(D/âM), at least M = Ω(n/D) labels per edge would be necessary.published_or_final_versio
Bipartite powers of k-chordal graphs
Let k be an integer and k \geq 3. A graph G is k-chordal if G does not have
an induced cycle of length greater than k. From the definition it is clear that
3-chordal graphs are precisely the class of chordal graphs. Duchet proved that,
for every positive integer m, if G^m is chordal then so is G^{m+2}.
Brandst\"adt et al. in [Andreas Brandst\"adt, Van Bang Le, and Thomas Szymczak.
Duchet-type theorems for powers of HHD-free graphs. Discrete Mathematics,
177(1-3):9-16, 1997.] showed that if G^m is k-chordal, then so is G^{m+2}.
Powering a bipartite graph does not preserve its bipartitedness. In order to
preserve the bipartitedness of a bipartite graph while powering Chandran et al.
introduced the notion of bipartite powering. This notion was introduced to aid
their study of boxicity of chordal bipartite graphs. Given a bipartite graph G
and an odd positive integer m, we define the graph G^{[m]} to be a bipartite
graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is
odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power
of G.
In this paper we show that, given a bipartite graph G, if G is k-chordal then
so is G^{[m]}, where k, m are positive integers such that k \geq 4 and m is
odd.Comment: 10 page
On the minimum and maximum selective graph coloring problems in some graph classes
Given a graph together with a partition of its vertex set, the minimum selective coloring problem consists of selecting one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is minimum. The contribution of this paper is twofold. First, we investigate the complexity status of the minimum selective coloring problem in some specific graph classes motivated by some models described in Demange et al. (2015). Second, we introduce a new problem that corresponds to the worst situation in the minimum selective coloring; the maximum selective coloring problem aims to select one vertex per partition set such that the chromatic number of the subgraph induced by the selected vertices is maximum. We motivat
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
The complexity of the characterization of networks supporting shortest-path interval routing
AbstractInterval Routing is a routing method that was proposed in order to reduce the size of the routing tables by using intervals and was extensively studied and implemented. Some variants of the original method were also defined and studied. The question of characterizing networks which support optimal (i.e., shortest path) Interval Routing has been thoroughly investigated for each of the variants and under different models, with only partial answers, both positive and negative, given so far. In this paper, we study the characterization problem under the most basic model (the one unit cost), and with the most restrictive memory requirements (one interval per edge). We prove that this problem is NP-hard (even for the restricted class of graphs of diameter at most 3). Our result holds for all variants of Interval Routing. It significantly extends some related NP-hardness result, and implies that, unless P=NP, partial characterization results of some classes of networks which support shortest path Interval Routing, cannot be pushed further to lead to efficient characterizations for these classes
Collective additive tree spanners for circle graphs and polygonal graphs
AbstractA graph G=(V,E) is said to admit a system of ÎŒ collective additive tree r-spanners if there is a system T(G) of at most ÎŒ spanning trees of G such that for any two vertices u,v of G a spanning tree TâT(G) exists such that the distance in T between u and v is at most r plus their distance in G. In this paper, we examine the problem of finding âsmallâ systems of collective additive tree r-spanners for small values of r on circle graphs and on polygonal graphs. Among other results, we show that every n-vertex circle graph admits a system of at most 2log32n collective additive tree 2-spanners and every n-vertex k-polygonal graph admits a system of at most 2log32k+7 collective additive tree 2-spanners. Moreover, we show that every n-vertex k-polygonal graph admits an additive (k+6)-spanner with at most 6nâ6 edges and every n-vertex 3-polygonal graph admits a system of at most three collective additive tree 2-spanners and an additive tree 6-spanner. All our collective tree spanners as well as all sparse spanners are constructible in polynomial time
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