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Given a planar graph G and a partition of the neighbors of each vertex v in four sets v↗, v↖, v↙, and v↘, the problem Windrose Planarity asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v↗v is above and to the right of v, (ii) each neighbor u ∈ v↖ is above and to the left of v, (iii) each neighbor u ∈ v↙ is below and to the left of v, (iv) each neighbor u ∈ v↘ is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow us to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is NP-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a given combinatorial embedding. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph with n vertices that has a windrose-planar drawing, we can construct one with at most one bend per edge and with at most 2n-5 bends in total, which lies on the 3n × 3n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area. [ABSTRACT FROM AUTHOR]

We study the following problem: given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G This problem fits the paradigm of extending a partial solution for a problem to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes an otherwise easy problem hard, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmas, which show that the planarity of partially embedded graphs exhibits the 'TONCAS' behavior "the obvious necessary conditions for planarity are also sufficient." These conditions are expressed in terms of the interplay between (1) the rotation system and containment relationships between cycles and (2) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we make our algorithm run in linear time. Finally, we consider several generalizations of the problem, such as minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. We also apply our algorithm to the simultaneous graph drawing problem SIMULTANEOUS EMBEDDING WITH FIXED EDGES (SEFE). There we obtain a linear-time algorithm for the case that one of the input graphs or the common graph has a fixed planar embedding. [ABSTRACT FROM AUTHOR]

Border Gateway Protocol (BGP) allows providers to express complex routing policies preserving high degrees of autonomy. However, unrestricted routing policies can adversely impact routing stability. A key concept to understand the interplay between autonomy and expressiveness on one side, and stability on the other side, is safety under filtering, i.e., guaranteed stability under autonomous usage of route filters. BGP route filters are used to selectively advertise specific routes to specific neighbors. In this paper, we provide a characterization of safety under filtering, filling the large gap between previously known necessary and sufficient conditions. Our characterization is based on the absence of a particular kind of dispute wheel, a structure involving circular dependencies among routing preferences. We exploit our result to show that networks admitting multiple stable states are provably unsafe under filtering, and the troublesome portion of the configuration can be pinpointed starting from the stable states alone. This is especially interesting from an operational point of view since networks with multiple stable states actually happen in practice (BGP wedgies). Finally, we show that adding filters to an existing configuration may lead to oscillations even if the configuration is safe under any link failure. Unexpectedly, we find policy configurations where misconfigured filters can do more harm than network faults. [ABSTRACT FROM PUBLISHER]

Abstract: The visualization of clustered graphs is a classical algorithmic topic that has several practical applications and is attracting increasing research interest. In this paper we deal with the visualization of clustered trees, a problem that is somehow foundational with respect to the one of visualizing a general clustered graph. We show many, in our opinion, surprising results that put in evidence how drawing clustered trees has many sharp differences with respect to drawing “plain” trees. We study a wide class of drawing standards, giving both negative and positive results. Namely, we show that there are clustered trees that do not have any drawing in certain standards and others that require exponential area. On the contrary, for many drawing conventions there are efficient algorithms that allow to draw clustered trees with polynomial asymptotically-optimal area. [Copyright &y& Elsevier]

We investigate the problem of computing the types of the relationships between Internet Autonomous Systems. We refer to the model introduced by Gao [IEEE/ACM TRANSACTIONS ON NETWORKING, 9(6):733-645, 2001] and Subramanian et al. (IEEE Infocom, 2002) that bases the discovery of such relationships on the analysis of the AS paths extracted from the BGP routing tables. We characterize the time complexity of the above problem, showing both NP-completeness results and efficient algorithms for solving specific cases. Motivated by the hardness of the general problem, we propose approximation algorithms and heuristics based on a novel paradigm and show their effectiveness against publicly available data sets. The experiments provide evidence that our algorithms perform significantly better than state-of-the-art heuristics. [ABSTRACT FROM AUTHOR]

In this paper we consider the important class of directed acyclic graphs (DAGs), and present the results of a comparative study on four popular drawing algorithms specifically developed for them. The study has been performed within a general experimental setting consisting of two large test suites of DAGs and a set of quality measures. The focus of the experiments has been the practical behavior of the algorithms with a geometric foundation compared to that of the algorithms with a topological foundation. The four algorithms exhibit various trade-offs with respect to the quality measures considered, and none of them clearly outperforms the others. Our analysis has motivated the development of a new hybrid strategy for drawing DAGs that performs quite well in practice. [ABSTRACT FROM AUTHOR]

A diagram is a drawing on the plane that represents a graph-like structure, where nodes are represented by symbols and edges are represented by curves connecting pairs of symbols. An automatic layout facility is a tool that receives as input a graph-like structure and is able to produce a diagram that nicely represents such a structure. Many systems use diagrams in the interaction with the users; thus, automatic layout facilities and algorithms for graphs layout have been extensively studied in the last years. We present a new approach in designing an automatic layout facility. Our approach is based on a modular management of a large collection of algorithms and on a strategy that, given the requirements of an application, selects a suitable algorithm for such requirements. The proposed approach has been used for designing the automatic layout facility of Diagram Server, a network server that offers to its clients several facilities for managing diagrams. [ABSTRACT FROM AUTHOR]

Internet Service Providers can enforce a fine-grained control of Interdomain Routing by cleverly configuring the Border Gateway Protocol. However, the price to pay for the flexibility of BGP is the lack of convergence guarantees. The literature on network protocol design introduced several sufficient conditions that routing policies should satisfy to guarantee convergence. However, a methodology to systematically check BGP policies for convergence is still missing. This paper presents two fundamental contributions. First, we describe a heuristic algorithm that statically checks BGP configurations for guaranteed routing convergence. Our algorithm has several highly desirable properties: i) it exceeds state-of-the-art algorithms by correctly reporting more configurations as stable, ii) it can be implemented efficiently enough to analyze Internet-scale configurations, iii) it is free from false positives, namely never reports a potentially oscillating configuration as stable, and iv) it can help spot troublesome points in a detected oscillation. Second, we propose an architecture for a modular tool that exploits our algorithm to process native router configurations and report the presence of potential oscillations. Such a tool can effectively integrate syntactic checkers and assist operators in verifying configurations. We validate our approach using a prototype implementation and show that it scales well enough to enable Internet-scale convergence checks. [ABSTRACT FROM PUBLISHER]