961 research outputs found

    The Internet's unexploited path diversity

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    The connectivity of the Internet at the Autonomous System level is influenced by the network operator policies implemented. These in turn impose a direction to the announcement of address advertisements and, consequently, to the paths that can be used to reach back such destinations. We propose to use directed graphs to properly represent how destinations propagate through the Internet and the number of arc-disjoint paths to quantify this network's path diversity. Moreover, in order to understand the effects that policies have on the connectivity of the Internet, numerical analyses of the resulting directed graphs were conducted. Results demonstrate that, even after policies have been applied, there is still path diversity which the Border Gateway Protocol cannot currently exploit.Comment: Submitted to IEEE Communications Letter

    On Directed Feedback Vertex Set parameterized by treewidth

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    We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time 2o(tlogt)nO(1)2^{o(t\log t)}\cdot n^{\mathcal{O}(1)} on general directed graphs, where tt is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time 2O(tlogt)nO(1)2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}. On the other hand, we show that if the input digraph is planar, then the running time can be improved to 2O(t)nO(1)2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}.Comment: 20

    Cuts in matchings of 3-connected cubic graphs

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    We discuss conjectures on Hamiltonicity in cubic graphs (Tait, Barnette, Tutte), on the dichromatic number of planar oriented graphs (Neumann-Lara), and on even graphs in digraphs whose contraction is strongly connected (Hochst\"attler). We show that all of them fit into the same framework related to cuts in matchings. This allows us to find a counterexample to the conjecture of Hochst\"attler and show that the conjecture of Neumann-Lara holds for all planar graphs on at most 26 vertices. Finally, we state a new conjecture on bipartite cubic oriented graphs, that naturally arises in this setting.Comment: 12 pages, 5 figures, 1 table. Improved expositio

    On Complexity of Minimum Leaf Out-branching Problem

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    Given a digraph DD, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in DD an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer kk, the problem of checking whether there is an out-branching with at most kk leaves is polynomial time solvable

    Sizing the length of complex networks

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    Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a small-world network and how does it compare to others' has remained a difficult challenge to answer. Here we establish a new synoptic representation that allows for a complete and accurate interpretation of the pathlength (and efficiency) of complex networks. We frame every network individually, based on how its length deviates from the shortest and the longest values it could possibly take. For that, we first had to uncover the upper and the lower limits for the pathlength and efficiency, which indeed depend on the specific number of nodes and links. These limits are given by families of singular configurations that we name as ultra-short and ultra-long networks. The representation here introduced frees network comparison from the need to rely on the choice of reference graph models (e.g., random graphs and ring lattices), a common practice that is prone to yield biased interpretations as we show. Application to empirical examples of three categories (neural, social and transportation) evidences that, while most real networks display a pathlength comparable to that of random graphs, when contrasted against the absolute boundaries, only the cortical connectomes prove to be ultra-short

    Minimum Cost Homomorphisms to Locally Semicomplete and Quasi-Transitive Digraphs

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    For digraphs GG and HH, a homomorphism of GG to HH is a mapping $f:\ V(G)\dom V(H)suchthat such that uv\in A(G)implies implies f(u)f(v)\in A(H).If,moreover,eachvertex. If, moreover, each vertex u \in V(G)isassociatedwithcosts is associated with costs c_i(u), i \in V(H),thenthecostofahomomorphism, then the cost of a homomorphism fis is \sum_{u\in V(G)}c_{f(u)}(u).Foreachfixeddigraph. For each fixed digraph H,theminimumcosthomomorphismproblemfor, the minimum cost homomorphism problem for H,denotedMinHOM(, denoted MinHOM(H),canbeformulatedasfollows:Givenaninputdigraph), can be formulated as follows: Given an input digraph G,togetherwithcosts, together with costs c_i(u),, u\in V(G),, i\in V(H),decidewhetherthereexistsahomomorphismof, decide whether there exists a homomorphism of Gto to H$ and, if one exists, to find one of minimum cost. Minimum cost homomorphism problems encompass (or are related to) many well studied optimization problems such as the minimum cost chromatic partition and repair analysis problems. We focus on the minimum cost homomorphism problem for locally semicomplete digraphs and quasi-transitive digraphs which are two well-known generalizations of tournaments. Using graph-theoretic characterization results for the two digraph classes, we obtain a full dichotomy classification of the complexity of minimum cost homomorphism problems for both classes
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