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Showing papers on "Path graph published in 2008"


Journal ArticleDOI
TL;DR: In this article, the diameter of a subgraph Γ 2 (R ) of a commutative ring with identity is characterized and the connectedness and diameter of this subgraph is analyzed.

101 citations


Journal ArticleDOI
TL;DR: A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G as mentioned in this paper, which is the state of the art.
Abstract: A G-design of order n is a decomposition of the complete graph on n vertices into edge-disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G-designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373-410, 2008

76 citations


Journal ArticleDOI
TL;DR: The maximum number of edges in a connected graph with n vertices if it contains no path with k+1 vertices is determined, and the extremal graphs are determined.

54 citations


Journal ArticleDOI
TL;DR: In this article, the authors characterize several classes of graphs for which E ǫ⩾ n is the energy of a graph on n vertices, where the energy is defined as ∑ i = 1 n | λ i |.

52 citations


Proceedings ArticleDOI
17 May 2008
TL;DR: It is proved that there exists a non-adaptive algorithm to find the edges of G using O(m log n / log m) queries of both types provided that m ≥ nε for any constant ε > 0 and it is shown that the same bound holds for all range of m.
Abstract: We consider the problem of finding an unknown graph by using two types of queries with an additive property. Given a graph, an additive query asks the number of edges in a set of vertices while a cross-additive query asks the number of edges crossing between two disjoint sets of vertices. The queries ask sum of weights for the weighted graphs. These types of queries were partially motivated in DNA shotgun sequencing and linkage discovery problem of artificial intelligence. For a given unknown weighted graph G with n vertices, m edges, and a certain mild condition on weights, we prove that there exists a non-adaptive algorithm to find the edges of G using O(m log n / log m) queries of both types provided that m ≥ ne for any constant e > 0. For an unweighted graph, it is shown that the same bound holds for all range of m. This settles a conjecture of Grebinski [23] for finding an unweighted graph using additive queries. We also consider the problem of finding the Fourier coefficients of a certain class of pseudo-Boolean functions. A similar coin weighing problem is also considered.

50 citations


Journal IssueDOI
TL;DR: In this article, King et al. showed that the conjecture holds if G is a quasi-line graph, extending a result of King et. al. who proved the conjecture for line graphs.
Abstract: A quasi-line graph is a graph in which the neighborhood of any vertex can be covered by two cliques; every line graph is a quasi-line graph. Reed conjectured that for any graph G, $\chi({{G}}) \leq\left \lceil {{{1}}\over {{2}}}(\Delta({{G}})+{{1}}+\omega({{G}}))\right\rceil$ [Reed, J Graph Theory 27 (1998), 177–212]. We prove that the conjecture holds if G is a quasi-line graph, extending a result of King et al. who proved the conjecture for line graphs [Eur J Comb 28 (2007), 2182–2187], and improving the bound of $\chi{{(}}{{G}}{{)}} \leq {3\over 2} \omega({{G}})$ given by Chudnovsky and Ovetsky [J Graph Theory 54 (2007), 41–50]. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 215–228, 2008

50 citations


Proceedings Article
08 Dec 2008
TL;DR: In practice, graphs may exhibit cluster structure; thus in the last part, this work presents a modified algorithm which achieves the "best of both worlds": it performs well locally in the presence of cluster structure, and globally on large diameter graphs.
Abstract: We continue our study of online prediction of the labelling of a graph. We show a fundamental limitation of Laplacian-based algorithms: if the graph has a large diameter then the number of mistakes made by such algorithms may be proportional to the square root of the number of vertices, even when tackling simple problems. We overcome this drawback by means of an efficient algorithm which achieves a logarithmic mistake bound. It is based on the notion of a spine, a path graph which provides a linear embedding of the original graph. In practice, graphs may exhibit cluster structure; thus in the last part, we present a modified algorithm which achieves the "best of both worlds": it performs well locally in the presence of cluster structure, and globally on large diameter graphs.

50 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that every graph on n vertices with at least cn r cliques on r vertices contains a complete r-partite subgraph with r−1 parts of size ⌊ c r log n⌋ and one part of size greater than n 1−cr−1.
Abstract: Let r≥2 and c>0. Every graph on n vertices with at least cn r cliques on r vertices contains a complete r-partite subgraph with r−1 parts of size ⌊ c r log n⌋ and one part of size greater than n 1−cr−1 . This result implies a quantitative form of the Erdos-Stone theorem.

48 citations


Proceedings Article
27 Jun 2008
TL;DR: The evaluation under the framework of SemEval-2007 WSI task shows the following: the approach produces less sense-conflating clusters than those produced by traditional graph-based approaches, and the approach outperforms the existing state-of-the-art results.
Abstract: Word Sense Induction (WSI) is the task of identifying the different senses (uses) of a target word in a given text. Traditional graph-based approaches create and then cluster a graph, in which each vertex corresponds to a word that co-occurs with the target word, and edges between vertices are weighted based on the co-occurrence frequency of their associated words. In contrast, in our approach each vertex corresponds to a collocation that co-occurs with the target word, and edges between vertices are weighted based on the co-occurrence frequency of their associated collocations. A smoothing technique is applied to identify more edges between vertices and the resulting graph is then clustered. Our evaluation under the framework of SemEval-2007 WSI task shows the following: (a) our approach produces less sense-conflating clusters than those produced by traditional graph-based approaches, (b) our approach outperforms the existing state-of-the-art results.

43 citations


Journal Article
TL;DR: In this paper, the relationship between the diameters and girths of the zero-divisor graphs of a subring of a commutative ring B and A was studied.
Abstract: The zero-divisor graph of a commutative ring R has the set of nonzero zero-divisors of R as its set of vertices, and two distinct vertices x and y are adjacent if and only if xy = 0. Let A be a subring of a commutative ring B. In this paper, we study the relationship between the diameters (resp., girths) of the zero-divisor graphs of A and B.

43 citations


Journal ArticleDOI
TL;DR: It is shown that for a path, the f"i"j's can be expressed as the products of fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers.

Proceedings Article
08 Dec 2008
TL;DR: This work presents a technique to compute a corresponding m × m Gram matrix of the pseudoinverse of the graph Laplacian in O(n + m2 + mS) time and discusses the application of this technique to fast label prediction on a generic graph.
Abstract: Given an n-vertex weighted tree with structural diameter S and a subset of m vertices, we present a technique to compute a corresponding m × m Gram matrix of the pseudoinverse of the graph Laplacian in O(n + m2 + mS) time. We discuss the application of this technique to fast label prediction on a generic graph. We approximate the graph with a spanning tree and then we predict with the kernel perceptron. We address the approximation of the graph with either a minimum spanning tree or a shortest path tree. The fast computation of the pseudoinverse enables us to address prediction problems on large graphs. We present experiments on two web-spam classification tasks, one of which includes a graph with 400,000 vertices and more than 10,000,000 edges. The results indicate that the accuracy of our technique is competitive with previous methods using the full graph information.

Journal ArticleDOI
TL;DR: In this paper, some rules for resistance distances of a graph G are established and it is shown that resistance distances between vertices in S depend only on the cardinality of N and the induced subgraph G[S].
Abstract: In this work, some rules for resistance distances of a graph G are established. Let S be a set of vertices of G such that all vertices in S have the same neighborhood N in G − S. If |S| = 2, 3, 4, simple formulae are derived to compute resistance distances between vertices in S in terms of the cardinality of N. These show that resistance distances between vertices in S depend only on the cardinality of N and the induced subgraph G[S]. One question arises naturally: does this property hold for S with arbitrarily many vertices? We answer this question by the following reduction principle: resistance distances between vertices in S can be computed as in the subgraph obtained from G[S N] by deleting all the edges between vertices in N.

Journal Article
TL;DR: TopoCluj as mentioned in this paper is a package for computing topological indices of the molecular graphs of nanostructures, which is based on the Wiener index, introduced by Wiener 1 as the half-sum of all topological distances in the hydrogendepleted graph representing the skeleton of the molecule.
Abstract: The nanostar dendrimer is part of a new group of macromolecules that appear to be photon funnels just like artificial antennas. It also shows good resistant to photo bleaching. The nanostar dendrimer promises great applications but first the structure and the energy transfer mechanism must be understood. Experimental and theoretical insight is needed in order to understand the energy transfer mechanism. Methodology Some algebraic definitions used for the study are given. Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. In a chemical graph, vertices represent atoms and edges represent bonds. These graphs have been used for affinity diagrams showing a relationship between chemical substances. Numbers reflecting certain structural features of a molecule that are obtained from its chemical graph are usually called topological indices. The Wiener index, W, one of widely used descriptors of molecular topology, was introduced in 1947 by Wiener 1 as the half-sum of all topological distances in the hydrogendepleted graph representing the skeleton of the molecule. Here, we denote by d(u,v), the topological distance between vertices u and v of the graph G, which is the length of a minimum path between these vertices. We encourage the readers to consult two survey articles by Dobrynin and his co-authors 2,3 and references therein for background material and historical aspect of Wiener index. Diudea 4-10 was the first scientist who investigated the mathematical properties of nanostructures. He and his team studied several nanostructures by computing their topological indices and designed a package named TopoCluj 11 for computing topological indices of the molecular graphs of nanostructures.

Journal ArticleDOI
TL;DR: It is proved that a connected graph of diameter at least 4 and of girth 7 or more (in particular, a tree) can be exactly reconstructed from metric balls of radius 2 of all its vertices.

Journal Article
TL;DR: In this paper, the authors studied the random walk system on the complete graph with n vertices and proved a Central Limit Theorem for the proportion of visited vertices at the end of the process.
Abstract: We study the following random walks system on the complete graph with n vertices. At time zero, there is a number of active and inactive particles living on the vertices. Active particles move as continuous-time, rate 1, random walks on the graph, and, any time a vertex with an inactive particle on it is visited, this particle turns into active and starts an independent random walk. However, for a fixed integer L 1, each active particle dies at the instant it reaches a total of L jumps without activating any particle. We prove a Law of Large Numbers and a Central Limit Theorem for the proportion of visited vertices at the end of the process.

Journal ArticleDOI
TL;DR: It is proved that h(G)=h"c (G)>=4 are obtained by substituting graphs into three consecutive vertices of a cycle; this yields a polynomial-time algorithm to check whether h"c(G))=@c" c(G).

Book ChapterDOI
26 May 2008
TL;DR: The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether thegraph is planar or not.
Abstract: For a graph G and a collection of vertex pairs {(s1, t1), ..., (sk, tk)}, the disjoint paths problem is to find k vertex-disjoint paths P1, ..., Pk, where Pi is a path from sii to ti for each i = 1, ..., k. This problem is one of the classic problems in combinatorial optimization and algorithmic graph theory, and has many applications, for example in transportation networks, VLSI layout, and recently, virtual circuits routing in high-speed networks. As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s1, t1), ..., (sk, tk)}. The objective is to find k paths P1, ..., Pk such that Pi is a path from si to ti and Pi and Pj have neither common vertices nor adjacent vertices for any distinct i, j. This problem setting is a generalization of the disjoint paths problem, since if we subdivide each edge, then desired disjoint paths would be induced disjoint paths. The problem is motivated by not only the disjoint paths problem but also the recognition of an induced subgraph. The latter has been developed in the recent years, and this is actually connected to the Strong Perfect Graph Theorem [4], and the recognition of the perfect graphs [2]. In this paper, we shall investigate the complexity issues of this problem. The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed planar graph, (ii) solvable in linear time when k is fixed and G is an undirected planar graph, (iii) NP-hard when k = 2 and G is an acyclic directed graph, (iv) NP-hard when k = 2 and G is an undirected general graph. (i) generalizes the result by Schrijver [22], while (ii) generalizes the result by Reed, Robertson, Schrijver and Seymour [17].

Proceedings Article
20 Jan 2008
TL;DR: In this paper, the authors show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n2).
Abstract: We show how to test whether a graph with n vertices and m edges is a partial cube, and if so how to find a distance-preserving embedding of the graph into a hypercube, in the near-optimal time bound O(n2), improving previous O(nm)-time solutions.

Journal ArticleDOI
TL;DR: This paper proves the conjecture that if G is a sufficiently large graph with n vertices and minimum degree at least(r-1r+@c)n, then G contains a copy of H, and generalises several results concerning sufficient degree conditions for the containment of spanning subgraphs.

Journal IssueDOI
TL;DR: In this article, it was shown that a graph on p vertices with q edges has at most (1 + ) 2.5 cycles, and if G is planar, then it has 2.
Abstract: Let G be a graph on p vertices with q edges and let r = q - p = 1. We show that G has at most ${15\over 16} 2^{r}$ cycles. We also show that if G is planar, then G has at most 2r - 1 = o(2r - 1) cycles. The planar result is best possible in the sense that any prism, that is, the Cartesian product of a cycle and a path with one edge, has more than 2r - 1 cycles. © Wiley Periodicals, Inc. J. Graph Theory 57: 255–264, 2008

Journal ArticleDOI
TL;DR: A classical theorem of Ore, providing sufficient conditional for a graph to be hamiltonian connected, is generalized to k^*-connected graphs.

Journal ArticleDOI
TL;DR: This framework to deter- mine optimal homogeneous networks is put to the test by applying it to the 2002 Jemaah Islamiyah Bali bombing and it is found that most aspects of this covert network can be explained by the theoretical framework.
Abstract: Covert organizations are constantly faced with a tradeoff between secrecy and operational effciency. Lindelauf, Borm and Hamers (2008) developed a theoretical framework to determine optimal homogeneous networks taking the above mentioned considerations explicitly into account. In this paper this framework is put to the test by applying it to the 2002 Jemaah Islamiyah Bali bombing. It is found that most aspects of this covert network can be explained by the theoretical framework. Some interactions however provide a higher risk to the network than others. The theoretical framework on covert networks is extended to accommodate for such heterogeneous interactions. Given a network structure the optimal location of one risky interaction is established. It is shown that the pair of individuals in the organization that should conduct the interaction that presents the highest risk to the organization, is the pair that is the least connected to the remainder of the network. Furthermore, optimal networks given a single risky interaction are approximated and compared. When choosing among a path, star and ring graph it is found that for low order graphs the path graph is best. When increasing the order of graphs under consideration a transition occurs such that the star graph becomes best. It is found that the higher the risk a single interaction presents to the covert network the later this transition from path to star graph occurs.

Journal ArticleDOI
TL;DR: In this article, the authors show that for a graph G of order n>=4, G is hamiltonian if and only if G is not isomorphic to any graph in {2K"1+K"2,K" 1+K1+k"2 + K" 3 + K 1+k 2 + K 3 + 2K"3 + K 4 + K 5 + K 6 + K 7 + K 8 + K 9 + K 10 + K 11 + K 12 + K 14 + K 15 + K 16 + K 17 +

Journal ArticleDOI
TL;DR: It is proved that, in an n-star graph, all fault-free vertices but at most two form a connected component, and that star graphs exhibit excellent fault-tolerant abilities in the sense that there exists a large functional network in a faulty star graph.
Abstract: In order to make a full evaluation of an interconnection network, it is essential to estimate the minimum size of a largest connected component of this network provided the faulty vertices in the network may break its connectedness. Star graphs are recognized as promising candidates for interconnection networks. This article addresses the size of a largest connected component of a faulty star graph. We prove that, in an n-star graph (n≥3) with up to 2n-4 faulty vertices, all fault-free vertices but at most two form a connected component. Moreover, all fault-free vertices but exactly two form a connected component if and only if the set of all faulty vertices is equal to the neighbourhood of a pair of fault-free adjacent vertices. These results show that star graphs exhibit excellent fault-tolerant abilities in the sense that there exists a large functional network in a faulty star graph.

Book ChapterDOI
07 Jul 2008
TL;DR: The first polynomial-time approximation scheme in planargraphs is given, considering a popular relaxation in which the solution is allowed to use multiple copies of the input-graph edges (paying separately for each copy).
Abstract: Consider the following problem: given a graph with edge-weightsand a subset Qof vertices, find a minimum-weight subgraphin which there are two edge-disjoint paths connecting every pair ofvertices in Q. The problem is a failure-resilient analogof the Steiner tree problem, and arises in telecommunicationsapplications. A more general formulation, also employed intelecommunications optimization, assigns a number (orrequirement) rve{0,1,2} to each vertex vin the graph; for each pairu,vof vertices, the solution network is requiredto contain min{ru,rv} edge-disjointu-to-vpaths. We address the problem in planar graphs, considering a popularrelaxation in which the solution is allowed to use multiple copiesof the input-graph edges (paying separately for each copy). Theproblem is SNP-hard in general graphs and NP-hard in planar graphs.We give the first polynomial-time approximation scheme in planargraphs. The running time is O(nlogn). Under the additional restriction that the requirements are in{0,2} for vertices on the boundary of a single face of a planargraph, we give a linear-time algorithm to find the optimalsolution.

Book ChapterDOI
15 Dec 2008
TL;DR: It is shown that the problem of augmenting G with an edge such that the resulting graph has minimum dilation can be considered and found in O(n 2logn) time, which solves an open problem of whether a linear-space algorithm with o(n 4) running time exists.
Abstract: Given a graph embedded in a metric space, its dilation is the maximum over all distinct pairs of vertices of the ratio between their distance in the graph and the metric distance between them. Given such a graph G with n vertices and m edges and consisting of at most two connected components, we consider the problem of augmenting G with an edge such that the resulting graph has minimum dilation. We show that we can find such an edge in $O((n^4\log n)/\sqrt m)$ time using linear space which solves an open problem of whether a linear-space algorithm with o(n 4) running time exists. We show that O(n 2logn) time is achievable if G is a simple path or the union of two vertex-disjoint simple paths. Finally, we show how to find an edge that maximizes the dilation of the resulting graph in O(n 3) time with O(n 2) space and in O(n 3logn) time with linear space.

Posted Content
TL;DR: This paper lists all graphs that are not path graphs and are minimal with this property by answering the characterizaton of path graphs by forbidden induced subgraphs question.
Abstract: A graph is a path graph if it is the intersection graph of a family of subpaths of a tree. In 1970, Renz asked for a characterizaton of path graphs by forbidden induced subgraphs. Here we answer this question by listing all graphs that are not path graphs and are minimal with this property.

Journal ArticleDOI
TL;DR: It is proved every 6-connected graph on n vertices with 5n-14 edges is 3-linked and this is optimal, in that there exist 6- connected graphs on n Vertices with5n-15 edges that are not 3- linked for arbitrarily large values of n.

Journal ArticleDOI
TL;DR: If G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number of vertices that must be removed from a graph G in order that the remaining subgraph have no component with more than k vertices is essentially the same.
Abstract: We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph have no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n → ∞, then the number in question is essentially the same for all values of k that satisfy both k → ∞ and k =o(n).