Showing papers on "Path graph published in 2006"
TL;DR: If both k and l are parameters, then (a, (b) and (d) are fixed-parameter tractable, while (c, and (e) are W[1]-hard.
323 citations
TL;DR: Evidence is provided that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth.
Abstract: We study the spreading dynamics on graphs with a power law degree distribution pkk � � , with 2 < �< 3, as an example of a branching process with a diverging reproductive number. We provide evidence that the divergence of the second moment of the degree distribution carries as a consequence a qualitative change in the growth pattern, deviating from the standard exponential growth. First, the population growth is extensive, meaning that the average number of vertices reached by the spreading process becomes of the order of the graph size in a time scale that vanishes in the large graph size limit. Second, the temporal evolution is governed by a polynomial growth, with a degree determined by the characteristic distance between vertices in the graph. These results open a path to further investigation on the dynamics on networks. unexpected based on previous mathematical studies. We also show that both the characteristic time separating the exponential and polynomial regimes and the polynomial degree depend on the characteristic distance between ver- tices. More important, in the limit of infinite graph sizes, the exponential regime is virtually absent, indicating that the polynomial regime is a novel and characteristic feature of the spreading dynamics on graphs with degree exponent 2 <�< 3 and, more generally, of branching processes with an unbounded average reproductive number. Consider a spreading process on a graph with a treelike structure. At t � 0, a vertex selected at random is infected by a ''virus,'' which can then propagate to other vertices through the graph edges. The causal tree representing the spreading process can be modeled as a branching process. Each vertex in the causal tree represents an infected vertex in the original graph, and each arc in the causal tree represents the generation of a secondary infected vertex from a primary infected vertex. The out-degree of a vertex in the causal tree gives the number of other vertices it infects, i.e., its reproductive number. In turn, the length of an arc A ! B in the causal tree gives the generation time, the time elapsed from the infection of the primary case A to the infection of the secondary case B. Finally, the vertex generation coincides with the topological distance from the first infected vertex, the root, in the original graph. We assume that the reproductive numbers are indepen- dent random variables with the probability distribution qd� k
128 citations
TL;DR: It is shown that deciding whether there is a planar straight-line embedding of G such that the vertices V are embedded onto the points P is NP-complete, even when G is 2-connected and 2-outerplanar.
Abstract: Let G = (V, E) be a graph with n vertices and let P be a set of n points in the plane. We show that deciding whether there is a planar straight-line embedding of G such that the vertices V are embedded onto the points P is NP-complete, even when G is 2-connected and 2-outerplanar. This settles an open problem posed in [2, 4, 13].
96 citations
TL;DR: A feedback vertex set of a graph is a set of vertices whose removal results in an acyclic graph and there is a cycle of length at most 6/ε (for ε ≥ 1/2, the authors can even improve this to just 6).
Abstract: A feedback vertex set (fvs) of a graph is a set of vertices whose removal results in an acyclic graph. We show that if an undirected graph on n vertices with minimum degree at least 3 has a fvs on at most 1/3n1 − e vertices, then there is a cycle of length at most 6/e (for e ≥ 1/2, we can even improve this to just 6).Using this, we obtain a O((12 log k/log log k p 6)knω algorithm for testing whether an undirected graph on n vertices has a fvs of size at most k. Here nω is the complexity of the best matrix multiplication algorithm. The previous best parameterized algorithm for this problem took O((2k p 1)kn2) time.We also investigate the fixed parameter complexity of weighted feedback vertex set problem in weighted undirected graphs.
86 citations
22 Jan 2006
TL;DR: This work shows how to compute in O(n4/3log 1/3n+n2/3k2/ 3log’n) time the distance between k given pairs of vertices of a planar graph G with n vertices, and speeds up previous algorithms for computing the dilation of geometric planar graphs.
Abstract: Let G be a planar graph with n vertices and non-negative edge-lengths. Given a set of k pairs of vertices, we are interested in computing the distance in G between those k pairs of vertices. We describe how this can be achieved in O(n2/3k2/3 log n + n4/3log1/3n) time, improving previous results for a large range of k. As possible applications, we show how this result speeds up previous algorithms for finding shortest non-contractible cycles for graphs on a bounded-genus surface or for computing the dilation of a geometric planar graph.
73 citations
TL;DR: The bound allows the Thomas–Wollan proof slightly to be modified slightly to show that every $2k-connected graph with average degree at least $12k$ is $k$-linked.
Abstract: A graph is $k$-linked if for every list of $2k$ vertices $\{s_1,{\ldots}\,s_k, t_1,{\ldots}\,t_k\}$, there exist internally disjoint paths $P_1,{\ldots}\, P_k$ such that each $P_i$ is an $s_i,t_i$-path. We consider degree conditions and connectivity conditions sufficient to force a graph to be $k$-linked.Let $D(n,k)$ be the minimum positive integer $d$ such that every $n$-vertex graph with minimum degree at least $d$ is $k$-linked and let $R(n,k)$ be the minimum positive integer $r$ such that every $n$-vertex graph in which the sum of degrees of each pair of non-adjacent vertices is at least $r$ is $k$-linked. The main result of the paper is finding the exact values of $D(n,k)$ and $R(n,k)$ for every $n$ and $k$.Thomas and Wollan [14] used the bound $D(n,k)\leq (n+3k)/2-2$ to give sufficient conditions for a graph to be $k$-linked in terms of connectivity. Our bound allows us to modify the Thomas–Wollan proof slightly to show that every $2k$-connected graph with average degree at least $12k$ is $k$-linked.
60 citations
TL;DR: In this paper, the zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∊ R\I|xy∊ I for some y∊ R/I} with distinct vertices x and y adjacent if and only if xy ∊ I.
Abstract: Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ I (R), is the graph whose vertices are the set {x ∊ R\I | xy ∊ I for some y ∊ R\I} with distinct vertices x and y adjacent if and only if xy ∊ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ I (R) ≅ Γ J (S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ I (R).
58 citations
TL;DR: The λ-number of a graph G, denoted by λ(G), is the minimum range of labels taken over all of its L(2, 1)-labelings, and it is shown that the λ of the Cartesian product of any two cycles is 6, 7 or 8.
40 citations
TL;DR: The (d, 1)-total number of vertices in a graph is the width of the smallest range of integers that suffices to label the vertices and the edges of a graph G so that no two adjacent vertices have the same color, no two incident edges has the same colour, and the distance between the color of a vertex and its incident edges is at least d.
Abstract: The (d,1)-total number $\lambda _{d}^{T}(G)$ of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same color, no two incident edges have the same color, and the distance between the color of a vertex and its incident edges is at least d. In this paper, we prove that $\lambda_{d}^{T}(G) \leq \Delta (G) + 2d - 2$ for connected graphs with a given maximum average degree. © 2005 Wiley Periodicals, Inc. J Graph Theory
38 citations
TL;DR: The proofs are based on Szemeredi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularities Lemma.
Abstract: Fix a 2-coloring Hk + 1 of the edges of a complete graph Kk + 1. Let C(n, Hk + 1) denote the maximum possible number of distinct edge-colorings of a simple graph on n vertices with two colors, which contain no copy of Kk + 1 colored exactly as Hk + 1. It is shown that for every fixed k and all n > n0(k), if in the colored graph Hk + 1 both colors were used, then C(n, Hk + 1) = 2tk(n), where tk(n) is the maximum possible number of edges of a graph on n vertices containing no K k + 1. The proofs are based on Szemeredi's Regularity Lemma together with the Simonovits Stability Theorem, and provide one of the growing number of examples of a precise result proved by applying the Regularity Lemma.
37 citations
TL;DR: In this article, it was shown that for every e>0, there exists a constant L such that every triangle-free graph on n vertices with minimum degree at least (1/3+e)n is homomorphic to a triangle free graph on at most L vertices.
Abstract: It is shown that for every e>0 there exists a constant L such that every triangle-free graph on n vertices with minimum degree at least (1/3+e)n is homomorphic to a triangle-free graph on at most L vertices.
04 Dec 2006
TL;DR: This paper deals with the graph G0 ⊕G1 obtained from merging two graphs G0 and G1, and shows that in a restricted HL-graph G of degree m (≥3), each pair of vertices are joined by a path in G \F of every length from 2m–3 to |V(G \F)|−1 for any set F of faulty elements (vertices and/or edges).
Abstract: In this paper, we deal with the graph G0 ⊕G1 obtained from merging two graphs G0 and G1 with n vertices each by n pairwise nonadjacent edges joining vertices in G0 and vertices in G1. The main problems studied are how fault-panconnectivity and fault-pancyclicity of G0 and G1 are translated into fault-panconnectivity and fault-pancyclicity of G0 ⊕G1, respectively. Applying our results to a subclass of hypercube-like interconnection networks called restricted HL-graphs, we show that in a restricted HL-graph G of degree m (≥3), each pair of vertices are joined by a path in G \F of every length from 2m–3 to |V(G \F)|−1 for any set F of faulty elements (vertices and/or edges) with |F| ≤m–3, and there exists a cycle of every length from 4 to |V(G \F)| for any fault set F with |F| ≤m–2.
TL;DR: In this article, it was shown that the proportion of vertices of degree k decreases at the rate of k - 3 at every step, and a strong law of large numbers for the maximum degree was proved.
TL;DR: It is proved that there exists a constant c2 such that if c > c2, then whp every graph which contains at least one edge from each ordered pair (ei,fi) has a component of size Ω(n).
Abstract: Let c be a constant and (e1,f1),(e2,f2),…,(ecn,fcn) be a sequence of ordered pairs of edges from the complete graph Kn chosen uniformly and independently at random. We prove that there exists a constant c2 such that if c > c2, then whp every graph which contains at least one edge from each ordered pair (ei,fi) has a component of size Ω(n), and, if c < c2, then whp there is a graph containing at least one edge from each pair that has no component with more than O(n1-e vertices, where e is a constant that depends on c2 - c. The constant c2 is roughly 0.97677. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006
Journal Article•
TL;DR: In this paper, the authors investigated the computational complexity of the extensibility problem of placing the remaining vertices to form a planar straight-line drawing of the whole graph and showed that it is NP-complete.
Abstract: We investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane, place the remaining vertices to form a planar straight-line drawing of the whole graph. We show that this extensibility problem, proposed in the 2003 Selected Open Problems in Graph Drawing [1], is NP-complete.
Journal Article•
TL;DR: Several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth are provided.
Abstract: The MULTICUT problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.
21 Jan 2006
TL;DR: In this paper, the authors provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.
Abstract: The Multicut problem is defined as: given an undirected graph and a collection of pairs of terminal vertices, find a minimum set of edges or vertices whose removal disconnects each pair. We mainly focus on the case of removing vertices, where we distinguish between allowing or disallowing the removal of terminal vertices. Complementing and refining previous results from the literature, we provide several NP-completeness and (fixed-parameter) tractability results for restricted classes of graphs such as trees, interval graphs, and graphs of bounded treewidth.
TL;DR: Among all connected cactuses with n vertices, this article found a unique graph whose largest eigenvalue (index, for short) is maximal among all cactus with n nodes.
Abstract: Among all connected cactuses with n vertices we find a unique graph whose largest eigenvalue (index, for short) is maximal.
TL;DR: The first large q-regular subgraphs appear discontinuously at an average vertex degree c3 − reg 3.3546 and contain immediately about 24% of all vertices in the graph.
Abstract: We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large q-regular subgraph, i.e., a subgraph with all vertices having degree equal to q. We reformulate this problem as a constraint-satisfaction problem, and solve it using the cavity method of statistical physics at zero temperature. For q = 3, we find that the first large q-regular subgraphs appear discontinuously at an average vertex degree c3 − reg 3.3546 and contain immediately about 24% of all vertices in the graph. This transition is extremely close to (but different from) the well-known 3-core percolation point c3 − core 3.3509. For q > 3, the q-regular subgraph percolation threshold is found to coincide with that of the q-core.
28 Jun 2006
TL;DR: The problems in this thesis can be explained and motivated using applications in the area of network design and analysis.
Abstract: In this thesis, three di®erent graph concepts are studied. A graph (V;E) consists of a set of vertices V and a set of edges E. Graphs are often used as a model for telecommunication networks, where the nodes of the network are represented by the vertices, and an edge is present between two vertices if the corresponding nodes are joined by a direct connection in the network. The two vertices joined by an edge are called its end vertices, and these two vertices are neighbors of each other. The degree of a vertex is its number of neighbors. The problems in this thesis can be explained and motivated using applications in the area of network design and analysis.
TL;DR: One of Erdos' favourite conjectures was that any triangle-free graph G on n vertices should contain a set of n/2 vertices that spans at most n2/50 edges, which is proved when the number of edges in G is either at mostn2/12 or at least n2-5.
13 Dec 2006
TL;DR: An O(c n ) time algorithm is given for these problems for any fixed constant r, where c is a positive constant strictly less than 2, solving a well known open problem and giving combinatorial lower bounds on the number of maximal r-regular induced subgraphs possible on a graph on n vertices and matching algorithmic upper bounds.
Abstract: Given a graph G = (V,E) on n vertices, the Maximum γ-Regular Induced Subgraph (M-γ-RIS) problems ask for a maximum sized subset of vertices R⊆V such that the induced subgraph on R, G[R], is γ-regular. We give an $\mathcal{O}(c^n)$ time algorithm for these problems for any fixed constant γ, where c is a positive constant strictly less than 2, solving a well known open problem. These algorithms are then generalized to solve counting and enumeration version of these problems in the same time. An interesting consequence of the enumeration algorithm is, that it shows that the number of maximal γ-regular induced subgraphs for a fixed constant γ on any graph on n vertices is upper bounded by o(2n).
We then give combinatorial lower bounds on the number of maximal γ-regular induced subgraphs possible on a graph on n vertices and also give matching algorithmic upper bounds.
We use the techniques and results obtained in the paper to obtain an improved exact algorithm for a special case of Induced Subgraph Isomorphism that is Induced γ-Regular Subgraph Isomorphism, where γ is a constant.
All the algorithms in the paper are simple but their analyses are not. Some of the upper bound proofs or algorithms require a new and different measure than the usual number of vertices or edges to measure the progress of the algorithm, and require solving an interesting system of polynomials.
TL;DR: This article presents several lower and upper bounds on Bm(n), and it is shown that modified Knodel graphs are m-message broadcast graphs for m ≤ min⌊log n⌋,n - 2⌆log n ⌋ and from the Cartesian product of some broadcast graphs better upper bounds are obtained.
Abstract: Multiple message broadcasting is the process of multiple message dissemination in a communication network in which m messages, originated by one vertex, are transmitted to all vertices of the network. A graph G with n vertices is called a m-message broadcast graph if its broadcast time is the theoretical minimum. Bm(n) is the minimum number of edges in any m-message broadcast graph on n vertices. An m-message minimum broadcast graph is a broadcast graph G on n vertices having Bm(n) edges. This article presents several lower and upper bounds on Bm(n). In particular, it is shown that modified Knodel graphs are m-message broadcast graphs for m ≤ min⌊log n⌋,n - 2⌊log n⌋. From the Cartesian product of some broadcast graphs we obtain better upper bounds on Bm(n), and in some cases we can prove that Bm(n) = O(n). The exact value of B2(2k) is also established. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 47(4), 218–224 2006
TL;DR: For a set X of vertices of a graph fulfilling local connectedness conditions, the existence of a cycle containing X is proved.
28 Aug 2006
TL;DR: An improved algorithm to compute the diameter is proposed and a sustainable parallel system with the Condor/MW framework is developed, and the diameters of 16- and 17-pancake graphs are computed by using PC clusters.
Abstract: An n-pancake graph is a graph whose vertices are the permutations of n symbols and each pair of vertices are connected with an edge if and only if the corresponding permutations can be transitive by a prefix reversal. Since the n-pancake graph has n! vertices, it is known to be a hard problem to compute its diameter by using an algorithm with the polynomial order of the number of vertices. Fundamental approaches of the diameter computation have been proposed. However, the computation of the diameter of 15-pancake graph has been the limit in practice. In order to compute the diameters of the larger pancake graphs, it is indispensable to establish a sustainable parallel system with enough scalability. Therefore, in this study, we have proposed an improved algorithm to compute the diameter and have developed a sustainable parallel system with the Condor/MW framework, and computed the diameters of 16- and 17-pancake graphs by using PC clusters.
TL;DR: This article considers the problems of finding heavy paths and heavy trees of k edges and shows that in every complete weighted graph on n vertices there exists a subgraph with approximately α/1−α2n edges that contains an α-approximate solution for every k = 1,…, n − 1.
Abstract: Consider a graph problem which is associated with a parameter, for example, that of finding a longest tour spanning k vertices. The following question is natural: Is there a small subgraph that contains an optimal or near optimal solution for every possible value of the given parameter? Such a subgraph is said to be robust. In this article we consider the problems of finding heavy paths and heavy trees of k edges. In these two cases, we prove surprising bounds on the size of a robust subgraph for a variety of approximation ratios. For both problems, we show that in every complete weighted graph on n vertices there exists a subgraph with approximately α/1−α2n edges that contains an α-approximate solution for every k = 1,…, n − 1. In the analysis of the tree problem, we also describe a new result regarding balanced decomposition of trees. In addition, we consider variants in which the subgraph itself is restricted to be a path or a tree. For these problems, we describe polynomial time algorithms and corresponding proofs of negative results.
TL;DR: In this article, the minimum Wiener index for graphs with a given number n of vertices and given (vertex or edge) connectivity k, having minimum W index is determined.
Abstract: The graphs with a given number n of vertices and given (vertex or edge) connectivity k, having minimum Wiener index are determined. In both cases this is Kk + (K1 U Kn-k-1), the graph obtained by connecting all vertices of the complete graph Kk with all vertices of the graph whose two components are Kn-k-1 and K1. AMS Mathematics Subject Classification (2000): 05C12, 05C40 05C35.
25 Jun 2006
TL;DR: Data-dependent bounds on the fraction of mislabeled vertices are derived, based on the number (or total weight) of edges between vertices differing in predicted label (i.e., the size of the cut).
Abstract: I consider the setting of transductive learning of vertex labels in graphs, in which a graph with n vertices is sampled according to some unknown distribution; there is a true labeling of the vertices such that each vertex is assigned to exactly one of k classes, but the labels of only some (random) subset of the vertices are revealed to the learner. The task is then to find a labeling of the remaining (unlabeled) vertices that agrees as much as possible with the true labeling. Several existing algorithms are based on the assumption that adjacent vertices are usually labeled the same. In order to better understand algorithms based on this assumption, I derive data-dependent bounds on the fraction of mislabeled vertices, based on the number (or total weight) of edges between vertices differing in predicted label (i.e., the size of the cut).
Dissertation•
01 Jan 2006TL;DR: The heuristic has a low time complexity and generates fast k-broadcast schemes in many network topologies and an improved lower bound on Bk(n), the minimum possible number of edges in a k-Broadcast graph on n vertices.
Abstract: Broadcasting is a fundamental information dissemination problem, wherein a message is sent from one vertex, the originator, to all other vertices in a graph. In k-broadcasting, an informed vertex can sends the message to at most k uninformed neighbors in each time unit. This thesis presents several algorithms to perform efficient k-broadcasting. The algorithm KBT generates the optimal k-broadcast scheme in trees, while the algorithm KBC finds the k-broadcast center of a given tree. This thesis presents an efficient heuristic for k-broadcasting. The heuristic has a low time complexity and generates fast k-broadcast schemes in many network topologies.
A k-broadcast graph G is a graph on n vertices where the k-broadcast time of G is c logk+1n c . Bk(n) stands for the minimum possible number of edges in a k-broadcast graph on n vertices. A k-broadcast graph on n vertices with Bk(n) edges is a minimum k-broadcast graph, which is denoted by k-mbg. This thesis presents several new k-mbg's and an improved lower bound on Bk(n).
Journal Article•
TL;DR: It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant.
Abstract: It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant. This bound, conjectured by Brass, cannot be improved, apart from the value of the constant. We strengthen and generalize this result to the case when the graph has a crossing-free drawing on an orientable surface of higher genus and there is no restriction on the degrees of the vertices.