Showing papers on "Path graph published in 2009"
TL;DR: This work confirms a conjecture by Michael Fellows that there is a polynomial-time kernelization for Cluster Editing that leads to a problem kernel with at most 6k vertices, and presents a cubic-time algorithm that finds a graph G and an integer k>=0 such that G can be transformed into a cluster graph by at most k edge modifications.
136 citations
TL;DR: The asymptotically sharp upper bound W"e(G)@?2^55^5n^5+O(n^9^/^2) for graphs of order n is proved.
91 citations
TL;DR: Facts show that the degree information on the adjacent vertices is powerful enough for random walks to achieve the optimum hitting time.
90 citations
TL;DR: It is shown that, for every l, if a graph contains no odd K"l-expansion then its chromatic number is O(llogl), and that given a graph and a subset S of its vertex set, either there are k vertex-disjoint odd paths with endpoints in S, or there is a set X of at most 2k-2 vertices such that every odd path with both ends in S contains a vertex in X.
78 citations
03 Jul 2009
TL;DR: This paper provides a one pass $\tilde{O}(n/\epsilon^{2})$ space algorithm that produces a sparsification that approximates each cut to a (1 + *** ) factor, and shows that $\Omega(n \log \frac1\ep silon)$ space is necessary for a onePass streaming algorithm to approximate the min-cut, improving upon the *** (n ) lower bound that arises from lower bounds for testing connectivity.
Abstract: Analyzing massive data sets has been one of the key motivations for studying streaming algorithms. In recent years, there has been significant progress in analysing distributions in a streaming setting, but the progress on graph problems has been limited. A main reason for this has been the existence of linear space lower bounds for even simple problems such as determining the connectedness of a graph. However, in many new scenarios that arise from social and other interaction networks, the number of vertices is significantly less than the number of edges. This has led to the formulation of the semi-streaming model where we assume that the space is (near) linear in the number of vertices (but not necessarily the edges), and the edges appear in an arbitrary (and possibly adversarial) order.
However there has been limited progress in analysing graph algorithms in this model. In this paper we focus on graph sparsification, which is one of the major building blocks in a variety of graph algorithms. Further, there has been a long history of (non-streaming) sampling algorithms that provide sparse graph approximations and it a natural question to ask: since the end result of the sparse approximation is a small (linear) space structure, can we achieve that using a small space, and in addition using a single pass over the data? The question is interesting from the standpoint of both theory and practice and we answer the question in the affirmative, by providing a one pass $\tilde{O}(n/\epsilon^{2})$ space algorithm that produces a sparsification that approximates each cut to a (1 + *** ) factor. We also show that $\Omega(n \log \frac1\epsilon)$ space is necessary for a one pass streaming algorithm to approximate the min-cut, improving upon the *** (n ) lower bound that arises from lower bounds for testing connectivity.
66 citations
TL;DR: It is proved that every graph with tree-width k>=3 and maximum degree @D>=1 has tree-partition-width at most 24k@D, and it is shown that this bound is within a constant factor of optimal.
Abstract: A tree-partition of a graph G is a proper partition of its vertex set into 'bags', such that identifying the vertices in each bag produces a forest. The width of a tree-partition is the maximum number of vertices in a bag. The tree-partition-width of G is the minimum width of a tree-partition of G. An anonymous referee of the paper [Guoli Ding, Bogdan Oporowski, Some results on tree decomposition of graphs, J. Graph Theory 20 (4) (1995) 481-499] proved that every graph with tree-width k>=3 and maximum degree @D>=1 has tree-partition-width at most 24k@D. We prove that this bound is within a constant factor of optimal. In particular, for all k>=3 and for all sufficiently large @D, we construct a graph with tree-width k, maximum degree @D, and tree-partition-width at least (18-@e)k@D. Moreover, we slightly improve the upper bound to 52(k+1)(72@D-1) without the restriction that k>=3.
58 citations
TL;DR: It is considered how continuous-time quantum walks can be used for graph matching, and the problem of measuring graph similarity is considered, which gives results that are intermediate between the most sophisticated iterative techniques available, and simpler less complex ones.
49 citations
TL;DR: This work presents the first truly subcubic algorithm for APBP in general dense graphs and gives a procedure for computing the (max,min)-product of two arbitrary matrices over R in O(n 2+w/3 ) O( n 2.792 ) time, where n is the number of vertices and w is the exponent for matrix multiplication over rings.
Abstract: In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all pairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In par- ticular, we give a procedure for computing the (max,min)-product of two arbitrary matrices over R ( {¥, ¥} in O(n 2+w/3 ) O(n 2.792 ) time, where n is the number of vertices and w is the exponent for matrix multiplication over rings. Max-min products can be used to compute the maximum bottleneck values for all pairs of vertices together with a "successor matrix" from which one can extract an explicit maximum bottleneck path for any pair of vertices in time linear in the length of the path.
49 citations
24 Jan 2009
TL;DR: This work provides a polynomial-time algorithm that reduces a given instance of H-packing to an equivalent instance with at most O(k |V(H)| 1) vertices.
Abstract: For a fixed connected graph H, we consider the NP-complete H-packing problem, where, given an undirected graph G and an integer k e 0, one has to decide whether there exist k vertex-disjoint copies of H in G. We give a problem kernel of O(k |V(H)| 1) vertices, that is, we provide a polynomial-time algorithm that reduces a given instance of H-packing to an equivalent instance with at most O(k |V(H)| 1) vertices. In particular, this result specialized to H being a triangle improves a problem kernel for Triangle Packing from O(k 3) vertices by Fellows et al. [WG 2004] to O(k 2) vertices.
49 citations
29 Aug 2009
TL;DR: This work analyzes how the contacts of each person are connected to each other and the positions they occupy in the neighborhood network and compares the intensity of their communications to their positions, finding that the two are notindependent.
Abstract: We present here a method for analyzing the neighborhoods of all the vertices in a large graph. We first give an algorithm for characterizing a simple undirected graph that relies on enumeration of small induced subgraphs. We make a step further in this direction by identifying not only subgraphs but also the positions occupied by the different vertices of the graph, being thus able to compute the roles played by the vertices of the graph. We apply this method to the neighborhood of each vertex in a 2.7M vertices, 6M edges mobile phone graph. We analyze how the contacts of each person are connected to each other and the positions they occupy in the neighborhood network. Then we compare the intensity of their communications (duration and frequency) to their positions, finding that the two are notindependent. We finally interpret and explain the results using social studies on phone communications.
42 citations
TL;DR: The DISCO algorithm as mentioned in this paper is a divide-and-conquer algorithm for graph realization with sparse and noisy short-range intervertex distances as inputs, which is applied to the problem of determining protein moleculer structure.
Abstract: We propose the DISCO algorithm for graph realization in $\mathbb{R}^d$, given sparse and noisy short-range intervertex distances as inputs. Our divide-and-conquer algorithm works as follows. When a group has a sufficiently small number of vertices, the basis step is to form a graph realization by solving a semidefinite program. The recursive step is to break a large group of vertices into two smaller groups with overlapping vertices. These two groups are solved recursively, and the subconfigurations are stitched together, using the overlapping atoms, to form a configuration for the larger group. At intermediate stages, the configurations are improved by gradient descent refinement. The algorithm is applied to the problem of determining protein moleculer structure. Tests are performed on molecules taken from the Protein Data Bank database. For each molecule, given 20-30% of the inter-atom distances less than 6A that are corrupted by a high level of noise, DISCO is able to reliably and efficiently reconstruct the conformation of large molecules. In particular, given 30% of distances with 20% multiplicative noise, a 13000-atom conformation problem is solved within an hour with a root mean square deviation of 1.6A.
12 Aug 2009
TL;DR: It is shown that the computational complexity of the LONGEST PATH CONTRACTIBILITY problem restricted to P l-free graphs jumps from being polynomially solvable to being NP-hard at l=?6, while this jump occurs at l =?
Abstract: The 2- Disjoint Connected Subgraphs problem asks if a given graph has two vertex-disjoint connected subgraphs containing pre-specified sets of vertices. We show that this problem is NP-complete even if one of the sets has cardinality 2. The Longest Path Contractibility problem asks for the largest integer *** for which an input graph can be contracted to the path P *** on *** vertices. We show that the computational complexity of the Longest Path Contractibility problem restricted to P *** -free graphs jumps from being polynomially solvable to being NP-hard at ***= 6, while this jump occurs at ***= 5 for the 2- Disjoint Connected Subgraphs problem. We also present an exact algorithm that solves the 2- Disjoint Connected Subgraphs problem faster than ${\cal O}^*(2^n)$ for any n -vertex P *** -free graph. For ***= 6, its running time is ${\cal O}^*(1.5790^n)$. We modify this algorithm to solve the Longest Path Contractibility problem for P 6 -free graphs in ${\cal O}^*(1.5790^n)$ time.
TL;DR: This is the first result that connectivity which is a linear function of k guarantees the Erdős-Pósa type result for parity-k-linked graphs.
Abstract: A graph G is k-linked if G has at least 2k vertices, and for any 2k vertices x 1,x 2, …, x k ,y 1,y 2, …, y k , G contains k pairwise disjoint paths P 1, …, P k such that P i joins x i and y i for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P 1, …, P k can be chosen such that the parities of their length are prescribed. Thomassen [22] was the first to prove the existence of a function f(k) such that every f(k)-connected graph is parity-k-linked if the deletion of any 4k-3 vertices leaves a nonbipartite graph.
In this paper, we will show that the above statement is still valid for 50k-connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdős-Posa type result for parity-k-linked graphs.
TL;DR: This paper proves the Loebl-Komlos-Sos conjecture that if a graph G on n vertices has at least n/2 vertices of degree at least k, then G contains as subgraphs all trees on k+1 vertices.
TL;DR: The path embedding problem in star graphs is considered and it is proved that paths of all even lengths from d ( x , y ) to n !
TL;DR: This paper studies the following problem and proves that there are instances of the problem which require at least k-3 bends on n-k edges and how to compute the output in O(n^2logn) time in a real-RAM model.
Abstract: Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph G^'@?G on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of G^'. We concentrate on trees and show how to compute the output in O(n^2logn) time in a real-RAM model and with at most n-k edges with at most [email protected]?k/[email protected]? bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least k-3 bends on n-k edges.
TL;DR: It is shown that there is an embedding in three-dimensional space of each assembly graph such that smoothing of vertices according to a given set of polygonal paths results in an unlinked graph.
05 Dec 2009
TL;DR: It is shown that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k has a 3k-vertex kernel.
Abstract: We present a polynomial time algorithm that for any graph G and integer k ? 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G R on at most 3k vertices and an integer k? such that G has a spanning tree with at least k internal vertices if and only if G R has a spanning tree with at least k? internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."
TL;DR: In this article, the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices was determined, and the unique graph with the minimal least eigenvalue among all graphs in U ( n, k ) was determined.
02 Dec 2009
TL;DR: A 2-approximation algorithm for the Bandwidth problem that takes worst-case $\mathcal{O}(1.9797^n)$ = 3^{0.6217 n})$ time and uses polynomial space and improves both the previous best 2- and 3- approximation algorithms of Cygan et al.
Abstract: The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b.
In this paper, we present a 2-approximation algorithm for the Bandwidth problem that takes worst-case $\mathcal{O}(1.9797^n)$ $= \mathcal{O}(3^{0.6217 n})$ time and uses polynomial space. This improves both the previous best 2- and 3-approximation algorithms of Cygan et al. which have an $\mathcal{O}^*(3^n)$ and $\mathcal{O}^*(2^n)$ worst-case time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident on vertices in the same bucket or on vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a simple divide-and-conquer strategy along with dynamic programming to achieve this improved time bound.
TL;DR: It is proved that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular, and it is shown that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into twoClique-Transversal sets.
Abstract: A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.
TL;DR: In this article, it was shown that if A is k-edge-connected in G and B is an A-bridge such that B is a tree and every vertex in V(B) - A has degree 3, then either A is (k - 1)-edge-connections in G - E(B), or A is connected in G-E(B).
Abstract: A set A of vertices of an undirected graph G is called k-edge-connected in G if for all pairs of distinct vertices a, b∈A, there exist k edge disjoint a, b-paths in G. An A-tree is a subtree of G containing A, and an A-bridge is a subgraph B of G which is either formed by a single edge with both end vertices in A or formed by the set of edges incident with the vertices of some component of G - A. It is proved that (i) if A is k·(e + 2)-edge-connected in G and every A-bridge has at most e vertices in V(G) - A or at most e + 2 vertices in A then there exist k edge disjoint A-trees, and that (ii) if A is k-edge-connected in G and B is an A-bridge such that B is a tree and every vertex in V(B) - A has degree 3 then either A is k-edge-connected in G - e for some e∈E(B) or A is (k - 1)-edge-connected in G - E(B). © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 188–198, 2009
TL;DR: It is proved that the n-dimensional burnt pancake graph Bn is super spanning connected if and only if n ≠ 2.
Abstract: Let u and v be any two distinct vertices of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w*-container if it contains all the vertices of G. A graph G is w*-connected if there exists a w*-container between any two distinct vertices. Let κ(G) be the connectivity of G. A graph G is super spanning connected if G is i*-connected for 1 ≤ i ≤ κ(G). In this paper, we prove that the n-dimensional burnt pancake graph Bn is super spanning connected if and only if n ≠ 2.
01 Jan 2009
TL;DR: In this article, it was shown that for non-star-like trees with n vertices and k pendent vertices, the trees with minimal energy are those with n ≥ 6 and trees with second-order minimal energy if n ≥ 8.
Abstract: The energy of a graph is defined as the sum of the absolute values of its eigenvalues. A tree is non-starlike if it has at least two vertices of degree greater than two. For 4 ≤ k ≤ n − 2, we determine, in the class of non-starlike trees with n vertices and k pendent vertices, the trees with minimal energy if n ≥ 6 and the trees with second– minimal energy if n ≥ 8.
05 Dec 2009
TL;DR: The result implies that the known $\Omega(\sqrt{\log n/\log\log n})$ lower bound on the time complexity for a constant or polylogarithmic approximation does not hold for graphs of bounded arboricity.
Abstract: We give a deterministic distributed approximation algorithm for the maximum matching problem in graphs of bounded arboricity. Specifically, given 0 < ?< 1 and a positive integer a, the algorithm finds a matching of size at least (1 ? ?)m(G), where m(G) is the size of the maximum matching in an n-vertex graph G with arboricity at most a. The algorithm runs in O(log* n) rounds in the message passing model and it is the first sublogarithmic algorithm for the problem on such a wide class of graphs. Moreover, the result implies that the known $\Omega(\sqrt{\log n/\log\log n})$ lower bound on the time complexity for a constant or polylogarithmic approximation does not hold for graphs of bounded arboricity.
TL;DR: The problem of finding the largest group or association of vertices that are dense (denoted as dense vertexlet) in a weighted scale-free graph is addressed and an O(n$^2$ log n) heuristic graph mining algorithm that produces an approximate solution is presented.
Abstract: An important problem of knowledge discovery that has recently evolved in various reallife networks is identifying the largest set of vertices that are functionally associated. The topology of many real-life networks shows scale-freeness, where the vertices of the underlying graph follow a power-law degree distribution. Moreover, the graphs corresponding to most of the real-life networks are weighted in nature. In this article, the problem of finding the largest group or association of vertices that are dense (denoted as dense vertexlet) in a weighted scale-free graph is addressed. Density quantifies the degree of similarity within a group of vertices in a graph. The density of a vertexlet is defined in a novel way that ensures significant participation of all the vertices within the vertexlet. It is established that the problem is NP-complete in nature. An upper bound on the order of the largest dense vertexlet of a weighted graph, with respect to certain density threshold value, is also derived. Finally, an O(n$^2$ log n) (n denotes the number of vertices in the graph) heuristic graph mining algorithm that produces an approximate solution for the problem is presented.
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TL;DR: The energy of a graph G is defined as the sum of the absolute values of its eigenvalues as discussed by the authors, where E(G) is defined by attaching two pendent edges to each of the end vertices.
Abstract: The energy E(G) of a graph G is defined as the sum of the absolute values of its eigenvalues Let S2 be the star of order 2 (or K2) and Q be the graph obtained from S2 by attaching two pendent edges to each of the end vertices �
TL;DR: It is shown that starting from a graph G whose I(G; x) has only real roots, one can build an infinite family of graphs, some being well-covered, whose independence polynomials have only real Roots (and, sometimes, are also palindromic).
Abstract: A stable set in a graph G is a set of pairwise non-adjacent vertices, and the stability number α(G) is the maximum size of a stable set in G. The independence polynomial of G is $$I(G; x) = s_{0}+s_{1}x+s_{2}x^{2}+\cdots+s_{\alpha}x^{\alpha},\alpha=\alpha(G),$$of graphs, a graph U is induced-universal for $${\mathcal{F}}$$if every graph in $${\mathcal{F}}$$is an induced subgraph of U. We give a construction for an induced-universal graph for the family of graphs on n vertices with degree at most r, which has $$Cn^{\lfloor (r+1)/2\rfloor}$$vertices and $$Dn^{2\lfloor (r+1)/2\rfloor -1}$$edges, where C and D are constants depending only on r. This construction is nearly optimal when r is even in that such an induced-universal graph must have at least cnr/2 vertices for some c depending only on r.
Our construction is explicit in that no probabilistic tools are needed to show that the graph exists or that a given graph is induced-universal. The construction also extends to multigraphs and directed graphs with bounded degree.
TL;DR: An Ω(n2.3693) lower bound is proved that every c-Ramsey graph with n vertices contains Ω (n5/2) induced subgraphs, any two of which differ either in the number of vertices or in theNumber of edges.
Abstract: An n-vertex graph G is c-Ramsey if it contains neither a complete nor an empty induced subgraph of size greater than c log n. Erdős, Faudree and Sos conjectured that every c-Ramsey graph with n vertices contains Ω(n5/2) induced subgraphs, any two of which differ either in the number of vertices or in the number of edges, i.e., the number of distinct pairs (|V(H)|, |E(H)|), as H ranges over all induced subgraphs of G, is Ω(n5/2). We prove an Ω(n2.3693) lower bound.
01 Jan 2009
TL;DR: In this article, the minimal energy of trees with a given number of pendent vertices has been characterized in Tn,t, the set of trees of order n with at least t vertices of degree two.
Abstract: The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of eigenvalues of the graph. It is well known that in the case of trees the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. Ye and Yuan [On the minimal energy of trees with a given number of pendent vertices, MATCH Commun. Math. Comput. Chem. 57 (2007) 193-201.] and Yu and Lv [Minimum energy on trees with k pendent vertices, Lin. Algebra Appl. 418 (2006) 625-633] independently characterized the trees with the minimal energy among the trees with a given number of pendent vertices (that is, vertices of degree one). Let Tn,t be the set of trees of order n with at least t vertices of degree two. In the present paper, we characterize the tree with minimal energy or Hosoya index in Tn,t.