An orthogonal drawing of a graph is an embedding of the graph in the plane such that each edge is representable as a chain of alternately horizontal and vertical line segments. This style of drawing finds applications...
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An orthogonal drawing of a graph is an embedding of the graph in the plane such that each edge is representable as a chain of alternately horizontal and vertical line segments. This style of drawing finds applications in areas such as optoelectronic systems, information visualization and VLSI circuits. We present orthogonal drawings of the Kronecker product of two cycles around vertex partitions of the graph into grids. In the process, we derive upper bounds on the crossing number of the graph. The resulting upper bounds are within a constant multiple of the lower bounds. Unlike the Cartesian product that is amenable to an inductive treatment, the Kronecker product entails a case-to-case analysis since the results depend heavily on the parameters corresponding to the lengths of the two cycles. (C) 2011 Elsevier Inc. All rights reserved.
The notions of gamma and beta-acyclicity are two classic generalizations of the acyclicity of graphs to hypergraphs. They satisfy the property that, if a hypergraph is gamma-acyclic then it is beta-acyclic, and the re...
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The notions of gamma and beta-acyclicity are two classic generalizations of the acyclicity of graphs to hypergraphs. They satisfy the property that, if a hypergraph is gamma-acyclic then it is beta-acyclic, and the reverse is false. We give some new properties concerning these notions. First we show that we can strictly insert another notion of acyclicity between them, namely the fact of having a join tree with disjoint branches. And if we add a condition on the existence of such a join tree, we obtain a notion equivalent to gamma-acyclicity. Then we present two characterizations, consisting in applying successively a small set of rules, deciding gamma and beta-acyclicity respectively. (C) 2012 Elsevier B.V. All rights reserved.
In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Sp...
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In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, let G be a graph with it vertices, triangles and let Delta be the maximum number of triangles an edge of G is contained in. Our randomized algorithm colors the vertices of G with N = 1/p colors uniformly at random, counts monochromatic triangles, i.e., triangles whose vertices have the same color, and scales that count appropriately. We show that if p >= max(Delta logn/t . root log n/t) then for any constant epsilon > 0 our unbiased estimate T is concentrated around its expectation, i.e., Pr{[T - E vertical bar T vertical bar] >= epsilon E vertical bar T vertical bar} = o(1). Finally, our algorithm is amenable to being parallelized. We present a simple MAPREDUCE implementation of our algorithm. (C) 2011 Elsevier B.V. All rights reserved.
In [N. Alon, M. Feldman, A.D. Procaccia, M. Tennenholtz, A note on competitive diffusion through social networks, Inform. Process. Lett. 110 (2010) 221-225], the authors introduced a game-theoretic model of diffusion ...
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In [N. Alon, M. Feldman, A.D. Procaccia, M. Tennenholtz, A note on competitive diffusion through social networks, Inform. Process. Lett. 110 (2010) 221-225], the authors introduced a game-theoretic model of diffusion process through a network. They showed a relation between the diameter of a given network and existence of pure Nash equilibria in the game. Theorem 1 of their paper says that a pure Nash equilibrium exists if the diameter is at most two. However, we have an example which does not admit a pure Nash equilibrium even if the diameter is two. Hence we correct the statement of Theorem 1 of their paper. (C) 2011 Elsevier B.V. All rights reserved.
Given a graph G = (V, E.) with a subset S subset of E of edges, the edge subset feedback edge set problem is to find a smallest set F of edges such that in G' = (V, E - F) no cycle contains an edge in S. We also d...
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Given a graph G = (V, E.) with a subset S subset of E of edges, the edge subset feedback edge set problem is to find a smallest set F of edges such that in G' = (V, E - F) no cycle contains an edge in S. We also define a restricted version of this problem, in which no edge in S is allowed to be selected into F to form a solution. In this paper, we give a linear-time algorithm for the edge subset feedback edge set problem, and show that the restricted version is NP-hard for fixed vertical bar S vertical bar >= 2 and fixed-parameter tractable with parameter being k = vertical bar F vertical bar. (C) 2011 Elsevier B.V. All rights reserved.
We consider the problem of computing non-trivial Nash equilibria in additive hedonic games with symmetric 0/1-utilities. Such a game can be represented by an undirected unweighted graph G(V, E) where a non-trivial Nas...
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We consider the problem of computing non-trivial Nash equilibria in additive hedonic games with symmetric 0/1-utilities. Such a game can be represented by an undirected unweighted graph G(V, E) where a non-trivial Nash equilibrium corresponds to a partition of V into at least two sets such that each node has at least as many neighbours in its own set compared to any other set in the partition. We show that computing such an equilibrium is NP-complete. On the other hand, we show that such an equilibrium is computable in polynomial time if (1) G is triangle free, (2)G contains no 4-cycles sharing and edge, and (3) G is not a star. If G is not a star with girth at least 5 we show how to compute a non-trivial equilibrium in 0 (n)-time. (C) 2012 Elsevier B.V. All rights reserved.
The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for hole-free graphs is unkno...
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The Maximum Weight Independent Set (MWIS) problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. The complexity of the MWIS problem for hole-free graphs is unknown. In this paper, we first prove that the MWIS problem for (hole, dart, gem)-free graphs can be solved in O(n(3))-time. By using this result, we prove that the MWIS problem for (hole, dart)-free graphs can be solved in O(n(4))-time. Though the MWIS problem for (hole, dart, gem)-free graphs is used as a subroutine, we also give the best known time bound for the solvability of the MWIS problem in (hole, dart, gem)-free graphs. (C) 2012 Elsevier B.V. All rights reserved.
The efficiency of a set S subset of V in a graph G = (V, E), is defined as epsilon(S) = vertical bar{v is an element of V - S: vertical bar N(v) boolean AND S vertical bar = }vertical bar 1;in other words, the efficie...
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The efficiency of a set S subset of V in a graph G = (V, E), is defined as epsilon(S) = vertical bar{v is an element of V - S: vertical bar N(v) boolean AND S vertical bar = }vertical bar 1;in other words, the efficiency of a set S equals the number of vertices in V - S that are adjacent to exactly one vertex in S. A set S is called optimally efficient if for every vertex v is an element of V - S, epsilon(S boolean OR {v}) <= epsilon(S), and for every vertex u is an element of S, epsilon(S - {u}) < epsilon(S). We present a polynomial time self-stabilizing algorithm for finding an optimally efficient set in an arbitrary graph. This algorithm is designed using the distance-2 self-stabilizing model of computation. (C) 2012 Elsevier B.V. All rights reserved.
Let G = (V, E) be any finite simple graph. A mapping phi : E -> vertical bar k vertical bar is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic ...
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Let G = (V, E) be any finite simple graph. A mapping phi : E -> vertical bar k vertical bar is called an acyclic edge k-colouring of G, if any two adjacent edges have different colours and there are no bichromatic cycles in G. The smallest number k of colours such that G has an acyclic edge k-colouring is called the acyclic chromatic index of G and is denoted by chi(a)'(G). Fiamcik proved that Delta(G) . (Delta(G) - 1) + 1 is an upper bound for the acyclic chromatic index of a graph G and conjectured that chi(a)' (G) <= Delta(G) + 2. In 1991, Alon et al. proved that chi(a)'(G) <= 64 Delta(G). This bound was later improved to 16 Delta(G) by Molloy and Reed. In this paper we completely determine the acyclic chromatic index of the class of fully subdivided graphs. Namely, we prove that if we subdivide all edges of a given graph H, then for the obtained graph G it holds chi(a)'(G) = Delta(C), provided Delta(G) >= 3 or G is acyclic. Our proof is constructive and hence yields a polynomial time algorithm. This result is an extension of a result obtained in 1984 by Fiamcik for subdivisions of cubic graphs and improves a recent result presented by Muthu et al. (C) 2012 Elsevier B.V. All rights reserved.
We present an automatic approach to the construction of BabelNet, a very large, wide-coverage multilingual semantic network. Key to our approach is the integration of lexicographic and encyclopedic knowledge from Word...
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We present an automatic approach to the construction of BabelNet, a very large, wide-coverage multilingual semantic network. Key to our approach is the integration of lexicographic and encyclopedic knowledge from WordNet and Wikipedia. In addition, Machine Translation is applied to enrich the resource with lexical information for all languages. We first conduct in vitro experiments on new and existing gold-standard datasets to show the high quality and coverage of BabelNet. We then show that our lexical resource can be used successfully to perform both monolingual and cross-lingual Word Sense Disambiguation: thanks to its wide lexical coverage and novel semantic relations, we are able to achieve state-of the-art results on three different SemEval evaluation tasks. (C) 2012 Published by Elsevier B.V.
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