Showing papers on "Path graph published in 2012"
TL;DR: This technical note provides necessary and sufficient conditions, based on simple rules from number theory, to characterize all and only the nodes from which the network system is reachable (respectively observable) and observability.
Abstract: In this technical note we investigate the reachability and observability properties of a network system, running a Laplacian based average consensus algorithm, when the communication graph is a path or a cycle. Specifically, we provide necessary and sufficient conditions, based on simple rules from number theory, to characterize all and only the nodes from which the network system is reachable (respectively observable). Interesting immediate corollaries of our results are: i) a path graph is reachable (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 reachable (observable) from any pair of nodes if and only if n is a prime number. For any set of control (observation) nodes, we provide a closed form expression for the (unreachable) unobservable eigenvalues and for the eigenvectors of the (unreachable) unobservable subsystem.
209 citations
TL;DR: In this paper, the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling are analyzed. And the necessary and sufficient conditions for correct graph selection over pair-wise binary random fields are derived.
Abstract: The problem of graphical model selection is to estimate the graph structure of a Markov random field given samples from it. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size and the number of edges k, and/or the maximal node degree d, are allowed to increase to infinity as a function of the sample size n. For pair-wise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class Gp,k of graphs on vertices with at most k edges, and over the class Gp,d of graphs on p vertices with maximum degree at most d. For the class Gp,k, we establish the existence of constants c and c' such that if n ; c' k2 log p. Similarly, for the class Gp,d, we exhibit constants c and c' such that for n ; c' d3 log p.
205 citations
04 Oct 2012
TL;DR: This paper defines generalized translation and modulation operators for signals on graphs, and uses these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis.
Abstract: The prevalence of signals on weighted graphs is increasing; however, because of the irregular structure of weighted graphs, classical signal processing techniques cannot be directly applied to signals on graphs. In this paper, we define generalized translation and modulation operators for signals on graphs, and use these operators to adapt the classical windowed Fourier transform to the graph setting, enabling vertex-frequency analysis. When we apply this transform to a signal with frequency components that vary along a path graph, the resulting spectrogram matches our intuition from classical discrete-time signal processing. Yet, our construction is fully generalized and can be applied to analyze signals on any undirected, connected, weighted graph.
128 citations
TL;DR: This work focuses on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions and concludes that the N-intertwined approximation is superior to the HMF approximation in regular graphs.
Abstract: We introduce the ?-susceptible-infected-susceptible (SIS) spreading model, which is taken as a benchmark for the comparison between the N-intertwined approximation and the Pastor-Satorras and Vespignani heterogeneous mean-field (HMF) approximation of the SIS model. The N-intertwined approximation, the HMF approximation, and the ?-SIS spreading model are compared for different graph types. We focus on the epidemic threshold and the steady-state fraction of infected nodes in networks with different degree distributions. Overall, the N-intertwined approximation is superior to the HMF approximation. The N-intertwined approximation is exactly the same as the HMF approximation in regular graphs. However, for some special graph types, such as the square lattice graph and the path graph, the two mean-field approximations are both very different from the ?-SIS spreading model.
99 citations
TL;DR: The r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and the corresponding optimal solutions are determined and classified.
Abstract: The r-component connectivity κ r (G) of the non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. So, κ2 is the usual connectivity. In this paper, we determine the r-component connectivity of the hypercube Q n for r=2, 3, …, n+1, and we classify all the corresponding optimal solutions.
87 citations
01 Feb 2012
TL;DR: A SBV-Cut algorithm is proposed which identifies a set of balance vertices that can be used to effectively and efficiently bisect a directed graph and which provides better vertex-cut based expansion and modularity scores than its competitors and works several orders more efficiently than constraint-minimization based approaches.
Abstract: Graphs are used for modeling a large spectrum of data from the web, to social connections between individuals, to concept maps and ontologies. As the number and complexities of graph based applications increase, rendering these graphs more compact, easier to understand, and navigate through are becoming crucial tasks. One approach to graph simplification is to partition the graph into smaller parts, so that instead of the whole graph, the partitions and their inter-connections need to be considered. Common approaches to graph partitioning involve identifying sets of edges (or edge-cuts) or vertices (or vertex-cuts) whose removal partitions the graph into the target number of disconnected components. While edge-cuts result in partitions that are vertex disjoint, in vertex-cuts the data vertices can serve as bridges between the resulting data partitions; consequently, vertex-cut based approaches are especially suitable when the vertices on the vertex-cut will be replicated on all relevant partitions. A significant challenge in vertex-cut based partitioning, however, is ensuring the balance of the resulting partitions while simultaneously minimizing the number of vertices that are cut (and thus replicated). In this paper, we propose a SBV-Cut algorithm which identifies a set of balance vertices that can be used to effectively and efficiently bisect a directed graph. The graph can then be further partitioned by a recursive application of structurally-balanced cuts to obtain a hierarchical partitioning of the graph. Experiments show that SBV-Cut provides better vertex-cut based expansion and modularity scores than its competitors and works several orders more efficiently than constraint-minimization based approaches.
87 citations
TL;DR: In particular, Lov\'{a}sz and Simonovits as mentioned in this paper showed that the best possible lower bound is given by the complete multipartite graph with ε n^2 edges in which all but one vertex class is of the same size while the remaining one may be smaller.
Abstract: Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs.
The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\'{a}sz and Simonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by the complete multipartite graph with $\gamma n^2$ edges in which all but one vertex class is of the same size while the remaining one may be smaller.
Their conjecture was recently resolved for $r=3$ by Razborov and for $r=4$ by Nikiforov. In this article, we prove the conjecture for all values of $r$.
81 citations
TL;DR: It is proved that if G=(V,E) is an undirected graph then the size of a contagious set is bounded by @?"v"@?"Vmin{1,kd(v)+1} (where d(v) is the degree of v).
63 citations
TL;DR: A generic approach to relate counting subgraphs to counting graph homomorphisms and combines known combinatorial bounds, ideas from succinct data structures, partition functions, and the color coding technique, results are obtained.
Abstract: We introduce a generic approach for counting subgraphs in a graph. The main idea is to relate counting subgraphs to counting graph homomorphisms. This approach provides new algorithms and unifies several well-known results in algorithms and combinatorics, including the recent algorithm of Bjorklund, Husfeldt, and Koivisto for computing the chromatic polynomial, the classical algorithm of Kohn et al. for counting Hamiltonian cycles, Ryser's formula for counting perfect matchings of a bipartite graph, and color-coding-based algorithms of Alon, Yuster, and Zwick. By combining our method with known combinatorial bounds, ideas from succinct data structures, partition functions, and the color coding technique, we obtain the following new results. The number of optimal bandwidth permutations of a graph on n vertices excluding a fixed graph as a minor can be computed in time $ 2^{n+o(n)} $, in particular, in time $\mathcal{O}(2^{n}n^3)$ for trees and in time $2^{n+\mathcal{O}(\sqrt{n})}$ for planar graphs. Counti...
60 citations
10 Sep 2012
TL;DR: An algorithm is given that uses O(n) queries to the adjacency matrix of an n-vertex graph to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected.
Abstract: We use span programs to develop quantum algorithms for several graph problems. We give an algorithm that uses $O(n \sqrt d)$ queries to the adjacency matrix of an n-vertex graph to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also give O(n)-query algorithms that decide if a graph contains as a subgraph a path, a star with two subdivided legs, or a subdivided claw. These algorithms can be implemented time efficiently and in logarithmic space. One of the main techniques is to modify the natural st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.
57 citations
05 Mar 2012
TL;DR: The advice complexity of this problem --- the amount of oracle information an online algorithm needs in order to make optimal choices --- is investigated and it is proved that precisely ⌈n/2 ⌉−1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order.
Abstract: In online graph coloring a graph is revealed to an online algorithm one vertex at a time, and the algorithm must color the vertices as they appear. This paper starts to investigate the advice complexity of this problem --- the amount of oracle information an online algorithm needs in order to make optimal choices. We also consider a more general problem --- a trade-off between online and offline graph coloring.
In the paper we prove that precisely ⌈n/2 ⌉−1 bits of advice are needed when the vertices on a path are presented for coloring in arbitrary order. The same holds in the more general case when just a subset of the vertices is colored online. However, the problem turns out to be non-trivial for the case where the online algorithm is guaranteed that the vertices it receives form a subset of a path and are presented in the order in which they lie on the path. For this variant we prove that its advice complexity is βn+O(logn) bits, where β≈0.406 is a fixed constant (we give its closed form). This suggests that the generalized problem will be challenging for more complex graph classes.
TL;DR: In this article, the authors studied the two induced subgraphs Z0(Γ(R)) and T0( ΔΔ(R) of a commutative ring with nonzero identity, with vertices Z(R)\{0} and R\{0}, respectively.
Abstract: Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zero-divisors. The total graph of R is the (undirected) graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z0(Γ(R)) and T0(Γ(R)) of T(Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z0(Γ(R)) and T0(Γ(R)) are connected and compute their diameter and girth. We also investigate zero-divisor paths and regular paths in T0(Γ(R)).
TL;DR: This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.
Abstract: We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices.
The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph K N contains a monochromatic copy of H. A famous result of Chvaatal, Rodl, Szemeredi and Trotter states that there exists a constant c(Δ) such that r(H) ? c(Δ)n for every graph H with n vertices and maximum degree Δ. The important open question is to determine the constant c(Δ). The best results, both due to Graham, Rodl and Rucinski, state that there are positive constants c and c? such that $$2^{c'\Delta } \leqslant c(\Delta ) \leqslant ^{c\Delta \log ^2 \Delta }$$ . We improve this upper bound, showing that there is a constant c for which c(Δ) ? 2 cΔlogΔ .
The induced Ramsey number r ind (H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd?s conjectured the existence of a constant c such that, for any graph H on n vertices, r ind (H) ? 2 cnlogn . We move a step closer to proving this conjecture, showing that r ind (H) ? 2 cnlogn . This improves upon an earlier result of Kohayakawa, Promel and Rodl by a factor of logn in the exponent.
TL;DR: It is shown that if G is a connected graph of order n ≥ 3 with no isolated vertex, then fd ( G ) ≤ n − 2, and an infinite family of connected graphs achieving equality in this bound is constructed.
19 Sep 2012
TL;DR: It is shown that there are efficient algorithms to test if a polygonal path is self-approaching inℝ2 and ℝ3, but it is NP-hard to testif a given graph drawing in �”3 has a self- Approaching uv-path.
Abstract: In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner.
We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3)constructing a self-approaching Steiner network connecting a given set of points.
We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in ℝ2 and ℝ3, but it is NP-hard to test if a given graph drawing in ℝ3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.
TL;DR: This paper provides the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs, and uses a similar, but essentially simple algorithm.
Abstract: Given a graph $G$, the longest path problem asks to compute a simple path of $G$ with the largest number of vertices. This problem is the most natural optimization version of the well-known and well-studied Hamiltonian path problem, and thus it is NP-hard on general graphs. However, in contrast to the Hamiltonian path problem, there are only a few restricted graph families, such as trees, and some small graph classes where polynomial algorithms for the longest path problem have been found. Recently it has been shown that this problem can be solved in polynomial time on interval graphs by applying dynamic programming to a characterizing ordering of the vertices of the given graph [K. Ioannidou, G. B. Mertzios, and S. D. Nikolopoulos, Algorithmica, 61 (2011), pp. 320--341], thus answering an open question. In the present paper, we provide the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs. Our algorithm uses a similar, but essentially simple...
TL;DR: This paper gives a linear-time algorithm for finding a longest path between any two given vertices in a rectangular grid graph.
TL;DR: If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic, and it is proved that every connected graph G≠K2 on n vertices is Weighted- ⌊3n/2⌋- Antimagic.
Abstract: Suppose G is a graph, k is a non-negative integer. We say G is k-antimagic if there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . We say G is weighted-k-antimagic if for any vertex weight function w: V→ℕ, there is an injection f: E→{1, 2, …, |E| + k} such that for any two distinct vertices u and v, . A well-known conjecture asserts that every connected graph G≠K2 is 0-antimagic. On the other hand, there are connected graphs G≠K2 which are not weighted-1-antimagic. It is unknown whether every connected graph G≠K2 is weighted-2-antimagic. In this paper, we prove that if G has a universal vertex, then G is weighted-2-antimagic. If G has a prime number of vertices and has a Hamiltonian path, then G is weighted-1-antimagic. We also prove that every connected graph G≠K2 on n vertices is weighted- ⌊3n/2⌋-antimagic. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.
TL;DR: In this paper, the authors studied the component structure of planar graphs and showed that there are two critical periods in the evolution of P (n;M) with respect to width n 2 = 3 and height n 3 = 5.
Abstract: Let P (n;M) be a graph chosen uniformly at random from the family of all labeled planar graphs with n vertices and M edges. In the paper we study the component structure of P (n;M). Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of P (n;M). The first one, of width ( n 2=3 ), is analogous to the phase transition observed in the standard random graph models and takes place for M = n=2 +O(n 2=3 ), when the largest complex component is formed. Then, for M = n + O(n 3=5 ), when the complex components covers nearly all vertices, the second critical period of width n 3=5 occurs. Starting from that moment increasing of M mostly affects the density of the complex components, not its size.
TL;DR: It is shown that a path-based tree support can be computed in polynomial time if it exists and is shown to have a monotone drawing.
TL;DR: In this article, the total graph of a commutative ring R with respect to its proper ideal I was studied, which is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y Z(R).
Abstract: Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T(). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y R, the vertices x and y are adjacent if and only if x + y S(I). The total graph of a commutative ring, that denoted by T(), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, ; this is an important result on the definition.
TL;DR: It is shown that for a fixed r, the number of maximal r-regular induced subgraphs in any graph with $n$ vertices is upper bounded by $\mathcal{O}(c^n)$, where $c$ is a positive constant strictly less than $2$.
Abstract: We show that for a fixed $r$, the number of maximal $r$-regular induced subgraphs in any graph with $n$ vertices is upper bounded by $\mathcal{O}(c^n)$, where $c$ is a positive constant strictly less than $2$. This bound generalizes the well-known result of Moon and Moser, who showed an upper bound of $3^{n/3}$ on the number of maximal independent sets of a graph on $n$ vertices. We complement this upper bound result by obtaining an almost tight lower bound on the number of (possible) maximal $r$-regular induced subgraphs possible in a graph on $n$ vertices. Our upper bound results are algorithmic. That is, we can enumerate all the maximal $r$-regular induced subgraphs in time $\mathcal{O}(c^n n^{\mathcal{O}(1)})$. A related question is that of finding a maximum-sized $r$-regular induced subgraph. Given a graph $G=(V,E)$ on $n$ vertices, the Maximum $r$-Regular Induced Subgraph (M-$r$-RIS) problem asks for a maximum-sized subset of vertices, $R \subseteq V$, such that the induced subgraph on $R$ is $r$-re...
Patent•
31 May 2012
TL;DR: In this paper, the authors present a method for efficient graph pattern matching and querying, which includes sending one or more first activation messages to a first set of vertices, wherein each of the first activation message has a value.
Abstract: Apparatuses, systems, and methods for efficient graph pattern matching and querying are disclosed. According to an aspect, a method includes providing a graph comprising vertices and edges associated with the vertices. Further, the method includes sending one or more first activation messages to a first set of the vertices, wherein each of the first activation messages has a value. The method also includes determining, at each vertex of the first set of vertices, whether the values of the one or more first activation messages received at the vertex meets a query condition. Further, the method also includes sending, at each vertex of the first set of vertices, one or more second activation messages from the vertex to a second set of vertices in response to determining that the values of the one or more first activation messages received at the vertex meets the query condition.
TL;DR: NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k is proved, and natural necessary extension properties of graphs of small convexity number are identified and studied.
Abstract: A set of vertices S in a graph is convex if it contains all vertices which belong to shortest paths between vertices in S. The convexity number c(G) of a graph G is the maximum cardinality of a convex set of vertices which does not contain all vertices of G. We prove NP-completeness of the problem to decide for a given bipartite graph G and an integer k whether c(G) ≥ k. Furthermore, we identify natural necessary extension properties of graphs of small convexity number and study the interplay between these properties and upper bounds on the convexity number.
04 Jul 2012
TL;DR: This work proves that the problem of testing whether a given AT-free graph contains some graph H as an induced topological minor admits a polynomial-time algorithm, as long as H is fixed; and shows that such an algorithm is essentially optimal.
Abstract: Paths P1,…,Pk in a graph G=(V,E) are said to be mutually induced if for any 1≤i
TL;DR: The first constant upper bound on the spanning ratio of this graph was given in this paper, using a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most the Euclidean distance between the vertices.
Abstract: Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most $\sqrt{50 + 22 \sqrt{5}}$ times the Euclidean distance between the vertices. We also give a lower bound on the spanning ratio of $\frac{1}{2}(11\sqrt{5} -17) \approx 3.798$.
TL;DR: In this paper, it was shown that for a graph G with diameter 2, the problem of computing the Rainbow Connection Number (RCN) is NP-hard and that rc(G)@?k+2 is the minimum integer k for which any two distinct vertices of G are connected.
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TL;DR: In this paper, the authors associate a graph Γ+R to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R.
Abstract: We associate a graph Γ+(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ+(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ+(R).
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TL;DR: In this paper, a structural characterization of the space of H-colorings of Z"m^d with fixed m and fixed n is given. But the analysis of the entropy of f is restricted to the case of n = 2 and H = n.
TL;DR: It is shown that the edges of every non-trivial graph can be weighted with integers 1, 2, 3, 4 in such a way that the adjacent vertices have different products of incident edge weights.