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Showing papers on "Path graph published in 2010"


Journal IssueDOI
TL;DR: In this paper, Caro et al. showed that if G has n vertices and minimum degree δ then rvc(G) <20n-δ, which is the smallest number of colors that are needed in order to make Grainbow edge-connected.
Abstract: An edge-colored graph Gis rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make Grainbow edge-connected. We prove that if Ghas nvertices and minimum degree δ then rc(G)<20n-δ. This solves open problems from Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), nR57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243–254). A vertex-colored graph Gis rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make Grainbow vertex-connected. One cannot upper-bound one of these parameters in terms of the other. Nevertheless, we prove that if Ghas nvertices and minimum degree δ then rvc(G)<11n-δ. We note that the proof in this case is different from the proof for the edge-colored case, and we cannot deduce one from the other. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 185–191, 2010

212 citations


Journal ArticleDOI
TL;DR: The Routing Betweenness Centrality (RBC) measure is defined that generalizes previously well known Betweenness measures such as the Shortest Path betweenness, Flow Betweenness, and Traffic Load Centrality by considering network flows created by arbitrary loop-free routing strategies.
Abstract: Betweenness-Centrality measure is often used in social and computer communication networks to estimate the potential monitoring and control capabilities a vertex may have on data flowing in the network. In this article, we define the Routing Betweenness Centrality (RBC) measure that generalizes previously well known Betweenness measures such as the Shortest Path Betweenness, Flow Betweenness, and Traffic Load Centrality by considering network flows created by arbitrary loop-free routing strategies.We present algorithms for computing RBC of all the individual vertices in the network and algorithms for computing the RBC of a given group of vertices, where the RBC of a group of vertices represents their potential to collaboratively monitor and control data flows in the network. Two types of collaborations are considered: (i) conjunctive—the group is a sequences of vertices controlling traffic where all members of the sequence process the traffic in the order defined by the sequence and (ii) disjunctive—the group is a set of vertices controlling traffic where at least one member of the set processes the traffic. The algorithms presented in this paper also take into consideration different sampling rates of network monitors, accommodate arbitrary communication patterns between the vertices (traffic matrices), and can be applied to groups consisting of vertices and/or edges.For the cases of routing strategies that depend on both the source and the target of the message, we present algorithms with time complexity of O(n2m) where n is the number of vertices in the network and m is the number of edges in the routing tree (or the routing directed acyclic graph (DAG) for the cases of multi-path routing strategies). The time complexity can be reduced by an order of n if we assume that the routing decisions depend solely on the target of the messages.Finally, we show that a preprocessing of O(n2m) time, supports computations of RBC of sequences in O(kn) time and computations of RBC of sets in O(n3n) time, where k in the number of vertices in the sequence or the set.

149 citations


Journal ArticleDOI
TL;DR: It is shown that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices, and this result is used to deduce that every string graphs with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges.
Abstract: A string graph is the intersection graph of a collection of continuous arcs in the plane. We show that any string graph with m edges can be separated into two parts of roughly equal size by the removal of $O(m^{3/4}\sqrt{\log m})$ vertices. This result is then used to deduce that every string graph with n vertices and no complete bipartite subgraph Kt,t has at most ctn edges, where ct is a constant depending only on t. Another application shows that locally tree-like string graphs are globally tree-like: for any e > 0, there is an integer g(e) such that every string graph with n vertices and girth at least g(e) has at most (1 + e)n edges. Furthermore, the number of such labelled graphs is at most (1 + e)nT(n), where T(n) = nn−2 is the number of labelled trees on n vertices.

83 citations


Journal IssueDOI
01 Mar 2010-Networks
TL;DR: In this paper, the authors consider a class of Cayley graphs that are generated by certain 3-cycles on the alternating group An and show that the remaining graph has a large connected component containing almost all vertices.
Abstract: In this article we consider a class of Cayley graphs that are generated by certain 3-cycles on the alternating group An. These graphs are generalizations of the alternating group graph AGn. We look at the case when the 3-cycles form a “tree-like structure,” and analyze its fault resiliency. We present a number of structural theorems and prove that even with linearly many vertices deleted, the remaining graph has a large connected component containing almost all vertices. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

53 citations


Book ChapterDOI
06 Jul 2010
TL;DR: An O(m+n)-time algorithm is presented that tests if a given directed graph is 2-vertex connected, and an O(n)- space data structure is designed that can compute in O(log2 n) time two internally vertex-disjoint paths from s to t.
Abstract: We present an O(m+n)-time algorithm that tests if a given directed graph is 2-vertex connected, where m is the number of arcs and n is the number of vertices. Based on this result we design an O(n)- space data structure that can compute in O(log2 n) time two internally vertex-disjoint paths from s to t, for any pair of query vertices s and t of a 2-vertex connected directed graph. The two paths can be reported in additional O(k) time, where k is their total length.

47 citations


Proceedings ArticleDOI
05 Jun 2010
TL;DR: This paper presents the first efficient connectivity oracle for graphs susceptible to vertex failures, and shows there is an ~O(m)-space oracle that processes any set of d failed edges in O(d2 log log n) time and, thereafter, answers connectivity queries in O-log log n time.
Abstract: Dynamic graph connectivity algorithms have been studied for many years, but typically in the most general possible setting, where the graph can evolve in completely arbitrary ways. In this paper we consider a dynamic subgraph model. We assume there is some fixed, underlying graph that can be preprocessed ahead of time. The graph is subject only to vertices and edges flipping "off" (failing) and "on" (recovering), where queries naturally apply to the subgraph on edges/vertices currently flipped on. This model fits most real world scenarios, where the topology of the graph in question (say a router network or road network) is constantly evolving due to temporary failures but never deviates too far from the ideal failure-free state.We present the first efficient connectivity oracle for graphs susceptible to vertex failures. Given vertices u and v and a set D of d failed vertices, we can determine if there is a path from u to v avoiding D in time polynomial in d log n. There is a tradeoff in our oracle between the space, which is roughly mne, for 0

42 citations


Journal ArticleDOI
TL;DR: For the special graph H on six vertices, the crossing numbers of its join with n isolated vertices as well as with the path P"n on n vertices and with the cycle C"n are given.

38 citations


Book ChapterDOI
09 Jun 2010
TL;DR: In this article, the integrality gap of the natural LP relaxation of the problem is Θ(logn), where n denotes the number of vertices in the graph, and the primal-dual method is used to obtain a constant factor approximation algorithm.
Abstract: We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge We obtain a constant-factor approximation algorithm, based on the primal-dual method Moreover, we show that the integrality gap of the natural LP relaxation of the problem is Θ(logn), where n denotes the number of vertices in the graph

36 citations


Journal ArticleDOI
TL;DR: It is proved that the surviving rate of every $n-vertex outerplanar graph is at least $1-\Theta(\frac{k^{2}\log n}{n})$, which is asymptotically tight.
Abstract: The firefighter problem is the following discrete-time game on a graph. Initially, a fire starts at a vertex of the graph. In each round, a firefighter protects one vertex not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate of a graph is the average percentage of vertices that can be saved when a fire starts randomly at one vertex of the graph, which measures the defense ability of a graph as a whole. In this paper, we study the surviving rates of graphs with bounded treewidth. We prove that the surviving rate of every $n$-vertex outerplanar graph is at least $1-\Theta(\frac{\log n}{n})$, which is asymptotically tight. We also prove that if $k$ firefighters are available in each round, then the surviving rate of an $n$-vertex graph with treewidth at most $k$ is at least $1-O(\frac{k^{2}\log n}{n})$. Furthermore, we show that the greedy strategy of Hartnell and Li [Congr. Numer., 145 (2000), pp. 187-192] for trees saves at least $1-\Theta(\frac{\log n}{n})$ percent of vertices on average for an $n$-vertex tree. Our results settle a conjecture and two problems of Cai and Wang [SIAM J. Discrete Math., 23 (2009), pp. 1814-1826] in affirmative.

32 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the maximum and minimum Zagreb indices for all connected graphs with n vertices and k cut edges are obtained at Knk (resp. Pnk), where Knk is a graph obtained by joining k independent vertices to one vertex of Kn−k and Pnkn is a vertex obtained by connecting a pendent path Pk+1 to one node of Cn−k.
Abstract: For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M1- and M2-value are obtained, respectively, and uniquely, at Knk (resp. Pnk), where Knk is a graph obtained by joining k independent vertices to one vertex of Kn−k and Pnk is a graph obtained by connecting a pendent path Pk+1 to one vertex of Cn−k.

30 citations


Journal ArticleDOI
TL;DR: It is shown that O(mlognlogm) queries are enough provided m>=(logn)^@a for a sufficiently large constant @a, which is best possible up to a constant factor if m=0.

Book ChapterDOI
06 Sep 2010
TL;DR: A novel approximation algorithm is developed for a maximum graph orientation problem, where given an undirected graph on n vertices and a collection of vertex pairs, the goal is to orient the edges of the graph so that a maximum number of pairs are connected by a directed path.
Abstract: The orientation of physical networks is a prime task in deciphering the signaling-regulatory circuitry of the cell. One manifestation of this computational task is as a maximum graph orientation problem, where given an undirected graph on n vertices and a collection of vertex pairs, the goal is to orient the edges of the graph so that a maximum number of pairs are connected by a directed path. We develop a novel approximation algorithm for this problem with a performance guarantee of O(log n/ log log n), improving on the current logarithmic approximation. In addition, motivated by interactions whose direction is pre-set, such as protein-DNA interactions, we extend our algorithm to handle mixed graphs, a major open problem posed by earlier work. In this setting, we show that a polylogarithmic approximation ratio is achievable under biologically-motivated assumptions on the sought paths.

Proceedings ArticleDOI
01 Dec 2010
TL;DR: This paper investigates the observability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a path or a cycle, and provides necessary and sufficient conditions to characterize all and only the nodes from which the network system is observable.
Abstract: In this paper we investigate the observability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a path or a cycle. More in detail, we provide necessary and sufficient conditions, based on simple algebraic rules from number theory, to characterize all and only the nodes from which the network system is observable. Interesting immediate corollaries of our results are: (i) a path graph is observable from any single node if and only if the number of nodes of the graph is a power of two, n = 2i; i ∈ ℕ, and (ii) a cycle is observable from any pair of observation nodes if and only if n is a prime number. For any set of observation nodes, we provide a closed form expression for the unobservable eigenvalues and for the eigenvectors of the unobservable subspace.

01 Jan 2010
TL;DR: In this paper, the minimum Kirchhoff index among all connected graphs with n vertices and k cut edges was given, and the extremal graph was characterized by the authors.
Abstract: The Kirchhoff index Kf(G) of a graph G is the sum of resistance distances between all pairs of vertices. In this paper, we give the minimum Kirchhoff index among all connected graphs with n vertices and k cut edges, and characterize the extremal graph.

Journal ArticleDOI
Yi Wang1, Yi-Zheng Fan1
TL;DR: In this article, the authors characterized the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtained a lower bound for the least Eigenvalue of a connected graph in terms of the number of cuts vertices.

Proceedings ArticleDOI
23 Oct 2010
TL;DR: It is shown that for any eps > 0, and positive integers k and q such that q >= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - eps)N vertices, it is NP-hard to find an independent set of N/q^{k+1} vertices.
Abstract: We show that for any eps > 0, and positive integers k and q such that q >= 2k + 1, given a graph on N vertices that has a q-colorable induced sub graph of (1 - eps)N vertices, it is NP-hard to find an independent set of N/q^{k+1} vertices. This substantially improves upon the work of Dinur et al. [DKPS] who gave a corresponding bound of N/q^2. Our result implies that for any positive integer k, given a graph that has an independent set of approx (2^k + 1)^{-1} fraction of vertices, it is NP-hard to find an independent set of (2k + 1)^{(k+1)} fraction of vertices. This improves on the previous work of Engebretsen and Holmerin [EH] who proved a gap of 2^{-k} vs 2^{-{k choose 2}}, which is best possible using techniques (including those of [EH]) based on the query efficient PCP of Samorodnitsky and Trevisan [3].

Journal ArticleDOI
TL;DR: In this paper, the concepts of super magic and super magic strength of a graph are introduced, where the super magic is defined as the minimum of all constants c′(f) where the minimum is taken over all super magic labeling of G and is denoted by sms(G).
Abstract: By G(p, q) we denote a graph having p vertices and q edges, by V and E the vertex set and edge set of G respectively. A graph G(p, q) is said to have an edge magic labeling (valuation) with the constant (magic number) c(f) if there exists a one-to-one and onto function f: V ∪ E → {1, 2, …., p + q} such that f(u)+f(v)+f(uv) = c(f) for all uv ∈ E. An edge magic labeling f of G is called a super magic labeling if f(E) ={1, 2, …., q}. In this paper the concepts of the super magic and super magic strength of a graph are introduced. The super magic strength (sms) of a graph G is defined as the minimum of all constants c′(f) where the minimum is taken over all super magic labeling of G and is denoted by sms(G). This minimum is defined only if the graph has at least one such super magic labeling. In this paper, the super magic strength of some well known graphs P2n, P2n+1, K1,n, Bn,n, , Pn2 and (2n + 1)P2, Cn and Wn are obtained, where Pn is a path on n vertices, K1,n is a star graph on n+1 vertices, n-bistar Bn,n is the graph obtained from two copies of K1,n by joining the centres of two copies of K1,n by an edge e, if e is subdivided then Bn,n becomes , (2n + 1) P2 is 2n + 1 disjoint copies of P2, Pn2 is a square graph of Pn. Cn is a cycle on n vertices and Wn = Cn + K1 is wheel on n + 1 vertices.

Proceedings ArticleDOI
17 Jan 2010
TL;DR: A faster algorithm and a much simpler proof of the correctness for the edge-disjoint paths problem in general graphs is given and the algorithm is faster than Robertson and Seymour’s.
Abstract: We consider the following well-known problem, which is called the edge-disjoint paths problem.Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G.Output: Edge-disjoint paths P1, P2,..., Pk in G such that Pi joins si and ti for i = 1, 2,..., k.Robertson and Seymour's graph minor project gives rise to an O(m3) algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project, spanning 23 papers and at least 500 pages proof.We give a faster algorithm and a simpler proof of the correctness for the edge-disjoint paths problem for any fixed k. Our results can be summarized as follows:1. If an input graph G is either 4-edge-connected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤ 3-edge-cuts. (iii) Excluding large clique minors.2. When an input graph G is either 4-edge-connected or Eulerian, the number of terminals k is allowed to be non-trivially superconstant number, up to k = O((log log log n)1/2-e) for any e > 0. Thus our hidden constant in this case is dramatically smaller than Robertson-Seymour's. In addition, if an input graph G is either 4-edge-connected planar or Eulerian planar, k is allowed to be O((log n)1/2-e) for any e > 0. The same thing holds for bounded genus graphs. Moreover, if an input graph is either 4-edge-connected H-minor-free or Eulerian H-minor-free for fixed graph H, k is allowed to be O((log log n)1/2-e) for any e > 0.3. We also give our own algorithm for the edge-disjoint paths problem in general graphs. We basically follow Robertson-Seymour's algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, the time complexity of our algorithm is O(n2), which is faster than Robertson and Seymour's.

Journal ArticleDOI
TL;DR: The kPC problem can be solved in linear time on the class of proper interval graphs, that is, in O(n+m) time on a proper interval graph on n vertices and m edges.

Journal ArticleDOI
TL;DR: This work attaches labels to vertices in such a way that this length can be determined from the labels of x,y and the vertices X, and constructs labels of size O(k2log 2n) for a graph with n vertices of tree-width or clique-width k.
Abstract: Given a graph G we consider the problem of preprocessing it so that given two vertices x,y and a set X of vertices, we can efficiently report the shortest path (or just its length) between x,y that avoids X. We attach labels to vertices in such a way that this length can be determined from the labels of x,y and the vertices X. For a graph with n vertices of tree-width or clique-width k, we construct labels of size O(k 2log 2 n). The constructions extend to directed graphs. The problem is motivated by routing in networks in case of failures or of routing policies which forbid certain paths.

Journal ArticleDOI
TL;DR: An algorithm is presented that untangles the cycle graph Cn while keeping Ω(n2/3) vertices fixed, and an upper bound on the number of fixed vertices in the worst case is presented.
Abstract: Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C n while keeping Ω(n 2/3) vertices fixed. For any connected graph G, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of G. One consequence is that every 3-connected planar graph has a drawing δ such that at most O((nlog n)2/3) vertices are fixed in every untangling of δ.

Journal ArticleDOI
TL;DR: An iterative approach to calculating the irregularity strength of a graph, and develops a new algorithm that determines the exact value s(T) for trees T in which every two vertices of degree not equal to two are at distance at least eight.

Journal ArticleDOI
TL;DR: This paper first investigates the least eigenvalue @l"n(G), then it presents two sharp bounds on the spread s(G) of G, which is the simplest connected graph with n vertices and n edges.
Abstract: Let G be a simple connected graph with n vertices and n edges which we call a unicyclic graph. In this paper, we first investigate the least eigenvalue @l"n(G), then we present two sharp bounds on the spread s(G) of G.

01 Jan 2010
TL;DR: In this paper, the authors prove a classification result for a hitherto unexplored class of graph C -algebras, allowing them to classify all graph C-algesas on finitely many vertices with a finite linear ideal lattice if all pair of vertices are connected by infinitely many edges.
Abstract: At the cost of restricting the nature of the involved K- groups, we prove a classification result for a hitherto unexplored class of graph C -algebras, allowing us to classify all graph C -algebras on finitely many vertices with a finite linear ideal lattice if all pair of vertices are connected by infinitely many edges if they are connected at all.

Book ChapterDOI
26 Jul 2010
TL;DR: This paper presents an algorithm for finding an optimal L(2, 1)-labeling of a graph with time complexity O*(3.5616n), which improves a previous best result: O* (3.8739n).
Abstract: L(2, 1)-labeling is graph labeling model where adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. In this paper we present an algorithm for finding an optimal L(2, 1)-labeling (i.e. an L(2, 1)-labeling in which largest label is the least possible) of a graph with time complexity O*(3.5616n), which improves a previous best result: O*(3.8739n).

Journal Article
TL;DR: In this paper, the minimum Kirchhoff index among all connected graphs with n vertices and k cut edges was given, and the extremal graph was characterized, where k is the number of vertices in the graph.
Abstract: The Kirchhoff index Kf (G) of a graph G is the sum of resistance distances between all pairs of vertices. In this paper, we give the minimum Kirchhoff index among all connected graphs with n vertices and k cut edges, and characterize the extremal graph.

Journal ArticleDOI
TL;DR: This paper shows that the graph W"n has k vertex-disjoint (S,T)-paths containing all vertices of W"N if and only if k=2^n^-^1 or the graphW"n-([email protected]?T) has a perfect matching.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d.
Abstract: We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8].

Journal ArticleDOI
TL;DR: In this paper, the authors discuss distance two labeling in the context of graph operations, where adjacent vertices receive labels which differ by at least k and vertices which are at distance two apart receive labels that differ by k.
Abstract: Let G = ( V ; E ) be a connected graph. For integers j ? k , L ( j ; k )-labeling of a graph G is an integer labeling of the vertices in V such that adjacent vertices receive integers which differ by at least j and vertices which are at distance two apart receive labels which differ by at least k . In this paper we discuss L (2; 1) labeling (or distance two labeling) in the context of some graph operations.

Journal ArticleDOI
TL;DR: It is proved that every even regular graph and every odd regular graph with a 1-factor are super (a,1)-edge-antimagic total graphs.