467 research outputs found

    Extensions of positive definite functions on amenable groups

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    Let SS be a subset of a amenable group GG such that eSe\in S and S1=SS^{-1}=S. The main result of the paper states that if the Cayley graph of GG with respect to SS has a certain combinatorial property, then every positive definite operator-valued function on SS can be extended to a positive definite function on GG. Several known extension results are obtained as a corollary. New applications are also presented

    EPG-representations with small grid-size

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    In an EPG-representation of a graph GG each vertex is represented by a path in the rectangular grid, and (v,w)(v,w) is an edge in GG if and only if the paths representing vv an ww share a grid-edge. Requiring paths representing edges to be x-monotone or, even stronger, both x- and y-monotone gives rise to three natural variants of EPG-representations, one where edges have no monotonicity requirements and two with the aforementioned monotonicity requirements. The focus of this paper is understanding how small a grid can be achieved for such EPG-representations with respect to various graph parameters. We show that there are mm-edge graphs that require a grid of area Ω(m)\Omega(m) in any variant of EPG-representations. Similarly there are pathwidth-kk graphs that require height Ω(k)\Omega(k) and area Ω(kn)\Omega(kn) in any variant of EPG-representations. We prove a matching upper bound of O(kn)O(kn) area for all pathwidth-kk graphs in the strongest model, the one where edges are required to be both x- and y-monotone. Thus in this strongest model, the result implies, for example, O(n)O(n), O(nlogn)O(n \log n) and O(n3/2)O(n^{3/2}) area bounds for bounded pathwidth graphs, bounded treewidth graphs and all classes of graphs that exclude a fixed minor, respectively. For the model with no restrictions on the monotonicity of the edges, stronger results can be achieved for some graph classes, for example an O(n)O(n) area bound for bounded treewidth graphs and O(nlog2n)O(n \log^2 n) bound for graphs of bounded genus.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

    Analysis of scale-free networks based on a threshold graph with intrinsic vertex weights

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    Many real networks are complex and have power-law vertex degree distribution, short diameter, and high clustering. We analyze the network model based on thresholding of the summed vertex weights, which belongs to the class of networks proposed by Caldarelli et al. (2002). Power-law degree distributions, particularly with the dynamically stable scaling exponent 2, realistic clustering, and short path lengths are produced for many types of weight distributions. Thresholding mechanisms can underlie a family of real complex networks that is characterized by cooperativeness and the baseline scaling exponent 2. It contrasts with the class of growth models with preferential attachment, which is marked by competitiveness and baseline scaling exponent 3.Comment: 5 figure

    On retracts, absolute retracts, and folds in cographs

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    Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the size of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

    On edge intersection graphs of paths with 2 bends

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    An EPG-representation of a graph G is a collection of paths in a grid, each corresponding to a single vertex of G, so that two vertices are adjacent if and only if their corresponding paths share infinitely many points. In this paper we focus on graphs admitting EPG-representations by paths with at most 2 bends. We show hardness of the recognition problem for this class of graphs, along with some subclasses. We also initiate the study of graphs representable by unaligned polylines, and by polylines, whose every segment is parallel to one of prescribed slopes. We show hardness of recognition and explore the trade-off between the number of bends and the number of slopes. © Springer-Verlag GmbH Germany 2016

    On the (non-)existence of polynomial kernels for Pl-free edge modification problems

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    Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l

    On the Threshold of Intractability

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    We study the computational complexity of the graph modification problems Threshold Editing and Chain Editing, adding and deleting as few edges as possible to transform the input into a threshold (or chain) graph. In this article, we show that both problems are NP-complete, resolving a conjecture by Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128, 2001). On the positive side, we show the problem admits a quadratic vertex kernel. Furthermore, we give a subexponential time parameterized algorithm solving Threshold Editing in 2O(klogk)+poly(n)2^{O(\surd k \log k)} + \text{poly}(n) time, making it one of relatively few natural problems in this complexity class on general graphs. These results are of broader interest to the field of social network analysis, where recent work of Brandes (ISAAC, 2014) posits that the minimum edit distance to a threshold graph gives a good measure of consistency for node centralities. Finally, we show that all our positive results extend to the related problem of Chain Editing, as well as the completion and deletion variants of both problems

    Rainbow domination and related problems on some classes of perfect graphs

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    Let kNk \in \mathbb{N} and let GG be a graph. A function f:V(G)2[k]f: V(G) \rightarrow 2^{[k]} is a rainbow function if, for every vertex xx with f(x)=f(x)=\emptyset, f(N(x))=[k]f(N(x)) =[k]. The rainbow domination number γkr(G)\gamma_{kr}(G) is the minimum of xV(G)f(x)\sum_{x \in V(G)} |f(x)| over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs
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