Ordering Of Vertices Research Articles

How simple robots benefit from looking back

We study the sensor and movement capabilities that simple robots need in order to create a map of an unknown polygon of size n, and to meet. We consider robots that can move from vertex to vertex, can backtrack movements, and see distant vertices in counter-clockwise order but have no means of visibly identifying them. We show that such robots can always solve the weak rendezvous problem and reconstruct the visibility graph, given an upper bound on n. Our results are tight: The strong rendezvous problem, in which robots need to gather at a common location, cannot be solved in general, and without a bound on n, not even n can be determined. In terms of mobile agents exploring a graph, our result implies that they can reconstruct any graph that is the visibility graph of a simple polygon. This is in contrast to the known result that the reconstruction of arbitrary graphs is impossible in general, even if n is known.

Recognizing digraphs of Kelly-width 2

Kelly-width is a parameter of digraphs recently proposed by Hunter and Kreutzer as a directed analogue of treewidth. We give an alternative characterization of digraphs of bounded Kelly-width in support of this analogy, and the first polynomial-time algorithm recognizing digraphs of Kelly-width 2. For an input digraph G = ( V , A ) the algorithm outputs a vertex ordering and a digraph H = ( V , B ) with A ⊆ B witnessing either that G has Kelly-width at most 2 or that G has Kelly-width at least 3, in time linear in H .

Hiding data: a high-capacity distortionless approach

The widespread use of 3D models has become an interest in steganography. In this paper, we present a novel data hiding method for 3D models. Based on the representation information, the key idea is to consider the vertex index as a message block. A message consists of three types of message blocks, namely, unique, repeated and 1-bit or 0-bit repetitions blocks. Three embedding methods, namely, vertex index embedding (VIE), dynamic-length bit-string mapping (DBM), and repeated bits embedding (RBE), each best for a respective type of message block are devised. All message vertices are then arranged in the light of proposed vertex order and output as the stego-model. The message block is extracted from the vertex of the stego-model by the order of vertex sequence and its index in the ordered vertex list. While integrating the proposed methods our scheme gains high capacity compared to existing techniques while preserving reasonable robustness. In addition, our method is efficient, 25 times faster than previous techniques. With high capacity, several novel applications like content annotation for large documents, 3D meta model for related resources, etc., becomes possible, making the 3D model as an excellent data container.

A parallel algorithm for generating bicompatible elimination orderings of proper interval graphs

In this paper, we first show how a certain ordering of vertices, called bicompatible elimination ordering (BCO), of a proper interval graph can be used to solve efficiently several problems in proper interval graphs. We, then, propose an NC parallel algorithm (i.e., polylogarithmic-time employing a polynomial number of processors) in SIMD CRCW PRAM (Single Instruction Stream Multiple Data Stream Concurrent Read Concurrent Write Parallel Random Access Machine) model of parallel computation to compute a BCO of a proper interval graph. To the best of our knowledge, this is the first NC parallel algorithm to compute a BCO of a proper interval graph.

A mixed 0-1 linear programming formulation for the exact solution of the minimum linear arrangement problem

We are concerned with the exact solution of a graph optimization problem known as minimum linear arrangement (MinLA). Define the length of each edge of a graph with respect to a linear ordering of the graph vertices. Then, the MinLA problem asks for a vertex ordering that minimizes the sum of edge lengths. MinLA has several practical applications and is NP-Hard. We present a mixed 0-1 linear programming formulation of the problem, which led to fast optimal solutions for dense graphs of sizes up to n = 23.

The SDD and SSD support structure for the ALICE Inner Tracking System

The ALICE Inner Tracking System is designed to provide the determination of the primary vertex and of the secondary vertices in order to reconstruct charm and hyperon decays, particle identification and tracking of low-momentum particles and to improve the momentum and angle measurements of the Time Projecting Chamber (TPC). The device consists of six layers of silicon detectors and has been installed into the ALICE's cavern of the LHC accelerator. In this paper we describe the mechanical structure of the instrument and its behaviour during and after the assembly procedure.

On economical set representations of graphs

Graphs and Algorithms In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of simplicial vertex and perfect elimination ordering.

A SYSTEMATIC SCAN FOR 7-COLOURINGS OF THE GRID

We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles through finite-size subsets of vertices in a deterministic order and updates the colours assigned to the vertices of each subset. The systematic scan Markov chain that we present cycles through subsets consisting of 2×2 sub-grids and updates the colours assigned to the vertices using a procedure known as heat-bath. We give a computer-assisted proof that this systematic scan Markov chain mixes in O( log n) scans, where n is the size of the rectangular sub-grid. We make use of a heuristic to compute required couplings of colourings of 2×2 sub-grids. This is the first time the mixing time of a systematic scan Markov chain on the grid has been shown to mix for less than 8 colours. We also give partial results that underline the challenges of proving rapid mixing of a systematic scan Markov chain for sampling 6-colourings of the grid by considering 2×3 and 3×3 sub-grids.

Minimum Degree Orderings

It is known that, given an edge-weighted graph, a maximum adjacency ordering (MA ordering) of vertices can find a special pair of vertices, called a pendent pair, and that a minimum cut in a graph can be found by repeatedly contracting a pendent pair, yielding one of the fastest and simplest minimum cut algorithms. In this paper, we provide another ordering of vertices, called a minimum degree ordering (MD ordering) as a new fundamental tool to analyze the structure of graphs. We prove that an MD ordering finds a different type of special pair of vertices, called a flat pair, which actually can be obtained as the last two vertices after repeatedly removing a vertex with the minimum degree. By contracting flat pairs, we can find not only a minimum cut but also all extreme subsets of a given graph. These results can be extended to the problem of finding extreme subsets in symmetric submodular set functions.

Mutually independent hamiltonian cycles of binary wrapped butterfly graphs

Effective utilization of communication resources is crucial for improving performance in multiprocessor/communication systems. In this paper, the mutually independent hamiltonicity is addressed for its effective utilization of resources on the binary wrapped butterfly graph. Let G be a graph with N vertices. A hamiltonian cycle C of G is represented by 〈 u 1 , u 2 , … , u N , u 1 〉 to emphasize the order of vertices on C . Two hamiltonian cycles of G , namely C 1 = 〈 u 1 , u 2 , … , u N , u 1 〉 and C 2 = 〈 v 1 , v 2 , … , v N , v 1 〉 , are said to be independent if u 1 = v 1 and u i ≠ v i for all 2 ≤ i ≤ N . A collection of m hamiltonian cycles C 1 , … , C m , starting from the same vertex, are m -mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonicity of a graph G , denoted by I H C ( G ) , is defined to be the maximum integer m such that, for each vertex u of G , there exists a set of m -mutually independent hamiltonian cycles starting from u . Let B F ( n ) denote the n -dimensional binary wrapped butterfly graph. Then we prove that I H C ( B F ( n ) ) = 4 for all n ≥ 3 .

ℓ 2 2 Spreading Metrics for Vertex Ordering Problems

We design approximation algorithms for the vertex ordering problems Minimum Linear Arrangement, Minimum Containing Interval Graph, and Minimum Storage-Time Product, achieving approximation factors of $O(\sqrt{\log n}\log\log n)$, $O(\sqrt{\log n}\log\log n)$, and $O(\sqrt{\log T}\log\log T)$, respectively, the last running in time polynomial in T (T being the sum of execution times). The technical contribution of our paper is to introduce “l 22 spreading metrics” (that can be computed by semidefinite programming) as relaxations for both undirected and directed “permutation metrics,” which are induced by permutations of {1,2,…,n}. The techniques introduced in the recent work of Arora, Rao and Vazirani (Proc. of 36th STOC, pp. 222–231, 2004) can be adapted to exploit the geometry of such l 22 spreading metrics, giving a powerful tool for the design of divide-and-conquer algorithms. In addition to their applications to approximation algorithms, the study of such l 22 spreading metrics as relaxations of permutation metrics is interesting in its own right. We show how our results imply that, in a certain sense we make precise, l 22 spreading metrics approximate permutation metrics on n points to a factor of $O(\sqrt{\log n}\log\log n)$.

THE P-CENTER PROBLEM ON FUZZY NETWORKS AND REDUCTION OF COST

Here we consider the p-center problem on dierent types of fuzzy networks. In particular, we are interested in the networks with interval and triangular fuzzy arc lengths and vertex-weights. A methodology to obtain the best satisfaction level of the decision maker who wishes to reduce the cost within the tolerance limits is proposed. Illustrative examples are provided. The p-center problem is a well-known facility location problem. It arises in the following way. Suppose some demand points are given by the vertices of a network, and the weight of each vertex represents the demand at that point, and there is a path between every pair of vertices in order to transport products from one vertex to another. The decision maker is to locate p locations within the network where the facilities are to be located such that the maximum cost to transport products to each demand point from the nearest facility is minimized. It is assumed that these costs are directly proportional to the distances to be covered and the quantities of product to be transported. If the facilities are considered to be located at the vertices only, the p-center problem can be defined as follows. Let G = (V,E) be a connected undirected network, where V = {v1,v2,...,vn} is the set of vertices of G and E is the set of edges. As G is connected, there exists a path between every pair of vertices. The distance d(vi,vj) between two vertices vi and vj is denoted by dij, the length of the shortest path joining the vertices vi and vj. The weight wi is associated with the vertex vi for all i = 1,2,...,n. Let us consider all the subsets of {1,2,...,n} with p number of elements. Let those subsets be A1,A2,...,Am. So for a given set covering Ai, the distance to be covered in order to serve the vertex vj is given by ij = min

A Unified View of Graph Searching

Graph searching is perhaps one of the simplest and most widely used tools in graph algorithms. Despite this, few theoretical results are known about the vertex orderings that can be produced by a specific search algorithm. A simple characterizing property, such as is known for LexBFS, can aid greatly in devising algorithms, writing proofs of correctness, and showing impossibility results. This paper unifies our view of graph search algorithms by showing simple, closely related characterizations of various well-known search paradigms, including BFS and DFS. Furthermore, these characterizations naturally lead to other search paradigms, namely, maximal neighborhood search and LexDFS.

Fixed-Parameter Complexity of Minimum Profile Problems

The profile of a graph is an integer-valued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NP-hard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n−1+k, considering k as the parameter; this is a parameterization above guaranteed value, since n−1 is a tight lower bound for the profile. We present two fixed-parameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest.

The heat kernel as the pagerank of a graph

The concept of pagerank was first started as a way for determining the ranking of Web pages by Web search engines. Based on relations in interconnected networks, pagerank has become a major tool for addressing fundamental problems arising in general graphs, especially for large information networks with hundreds of thousands of nodes. A notable notion of pagerank, introduced by Brin and Page and denoted by PageRank, is based on random walks as a geometric sum. In this paper, we consider a notion of pagerank that is based on the (discrete) heat kernel and can be expressed as an exponential sum of random walks. The heat kernel satisfies the heat equation and can be used to analyze many useful properties of random walks in a graph. A local Cheeger inequality is established, which implies that, by focusing on cuts determined by linear orderings of vertices using the heat kernel pageranks, the resulting partition is within a quadratic factor of the optimum. This is true, even if we restrict the volume of the small part separated by the cut to be close to some specified target value. This leads to a graph partitioning algorithm for which the running time is proportional to the size of the targeted volume (instead of the size of the whole graph).

Corrections of proofs for Hansen and Mélot's two theorems

The Randić index R ( G ) of a (chemical) graph G is also called connectivity index. Hansen and Mélot [Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (2003) 1–14] in their paper, characterized the chemical trees of given order and number of pendent vertices which have the minimum and maximum Randić index, respectively. In this note, we point out the mistakes in the proofs of their results Theorems 8 and 10, while we still believe that the two theorems are true, and then we give their corrected proofs.

Structure of higher spin gauge interactions

In a previous paper, higher spin gauge field theory was formulated in an abstract way, essentially only keeping enough machinery to discuss gauge invariance of an action. The approach could be thought of as providing an interface (or syntax) toward an implementation (or semantics) yet to be constructed. The structure then revealed turned out to be that of a strongly homotopy Lie algebra. In the present paper, the framework will be connected to more conventional field theoretic concepts. The Fock complex vertex operator implementation of the interactions in the BRST-BV formulation of the theory will be elaborated. The relation between the vertex order expansion and homological perturbation theory will be clarified. A formal nonobstruction argument is reviewed. The syntactically derived sh-Lie algebra structure is semantically mapped to the Fock complex implementation, and it is shown that the equations governing the higher order vertices are reproduced. Global symmetries and subsidiary conditions are discussed and as a result the tracelessness constraints are discarded. Thus, all equations needed to compute the vertices to any order are collected. The framework is general enough to encompass all possible interaction terms. Finally, the abstract framework itself will be strengthened by showing that it can be naturally phrased in terms of category theory.

Ternary subdivision for quadrilateral meshes

A well-documented problem of Catmull and Clark subdivision surfaces is that, in the neighborhood of extraordinary points, the curvature is unbounded and fluctuates. In fact, since one of the eigenvalues that determines elliptic shape is too small, the limit surface can have a saddle point when the designer's input mesh suggests a convex shape. Here, we replace, near the extraordinary point, Catmull–Clark subdivision by another set of rules based on refining each bi-cubic B-spline into nine. This provides many localized degrees of freedom for special rules that need not reach out to neighbor vertices in order to tune the behavior. In this paper, we provide a strategy for setting such degrees of freedom and exhibit tuned ternary quad subdivision that yields surfaces with bounded curvature, nonnegative weights and full contribution of elliptic and hyperbolic shape components.

Further results on the enumeration of hamilton paths in Cayley digraphs on semidirect products of cyclic groups

First, let m and n be positive integers such that n is odd and gcd ( m , n ) = 1 . Let G be the semidirect product of cyclic groups given by G = Z 8 m ⋊ Z 2 n = 〈 x , y : x 8 m = 1 , y 2 n = 1 , and yxy - 1 = x 4 m + 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is one fewer than the number of visible lattice points that lie on the closed quadrilateral whose vertices in consecutive order are ( 0 , 0 ) , ( 4 mn 2 + 2 n , 16 m 2 n ) , ( n , 4 m ) , and ( 0 , 8 m ) . Second, let m and n be positive integers such that n is odd. Let G be the semidirect product of cyclic groups given by G = Z 4 m ⋊ Z 2 n = 〈 x , y : x 4 m = 1 , y 2 n = 1 , and yxy - 1 = x 2 m - 1 〉 . Then the number of hamilton paths in Cay ( G : x , y ) (with initial vertex 1) is ( 3 m - 1 ) n + m ⌊ ( n + 1 ) / 3 ⌋ + 1 .

Queue Layout of Bipartite Graph Subdivisions

For an integer d > 0, a d-queue layout of a graph consists of a total order of the vertices, and a partition of the edges into d sets of non-nested edges with respect to the vertex ordering. Recently V. Dujmovic and D. R. Wood showed that for every integer d ≥ 2, every graph G has a d-queue layout of a subdivision of G with 2⌈logdqn(G)⌉ + 1 division vertices per edge, where qn(G) is the queue number of G. This paper improves the result for the case of a bipartite graph, and shows that for every integer d ≥ 2, every bipartite graph Gm,n has a d-queue layout of a subdivision of Gm,n with ⌈logdn⌉-1 division vertices per edge, where m and n are numbers of vertices of the partite sets of Gm,n (m ≥ n).