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Harmful Internet hijacking incidents put in evidence how fragile interdomain routing is. In particular, the Border Gateway Protocol (BGP), which is used to exchange routing information between Internet entities, called Autonomous Systems (ASes), proved to be prone to attacks launched by a single malicious AS. Recent research contributions pointed out that even S-BGP, the secure variant of BGP that is being deployed, is not fully able to blunt traffic attraction attacks. Given a traffic flow between two ASes, we study how difficult it is for a malicious AS to devise a strategy for hijacking or intercepting that flow. The goal of the attack is to attract a traffic flow towards the malicious AS. While in the hijacking attack connectivity between the endpoints of a flow can be disrupted, in the interception attack connectivity must be maintained. We show that this problem marks a sharp difference between BGP and S-BGP. Namely, while it is solvable, under reasonable assumptions, in polynomial time for the type of attacks that are usually performed in BGP, it is NP-hard for S-BGP. Our study has several by-products. E.g., we solve a problem left open in the literature, stating when performing a hijacking in S-BGP is equivalent to performing an interception. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the time complexity of the problem Simultaneous Embedding with Fixed Edges (Sefe), that takes two planar graphs and as input and asks whether a planar drawing of and a planar drawing of exist such that: (i) each vertex is mapped to the same point in and in ; (ii) every edge is mapped to the same Jordan curve in and . First, we give a linear-time algorithm for Sefe when the intersection graph of and , that is the planar graph , is biconnected. Second, we show that Sefe, when is connected, is equivalent to a suitably-defined book embedding problem. Based on this equivalence and on recent results by Hong and Nagamochi, we show a linear-time algorithm for the Sefe problem when is a star. [Copyright &y& Elsevier]

In this paper we settle the computational complexity of two open problems related to the extension of the notion of level planarity to surfaces different from the plane. Namely, we show that the problems of testing the existence of a level embedding of a level graph on the surface of the rolling cylinder or on the surface of the torus, respectively known by the name of Cyclic Level Planarity and Torus Level Planarity , are polynomial-time solvable. Moreover, we show a complexity dichotomy for testing the Simultaneous Level Planarity of a set of level graphs, with respect to both the number of level graphs and the number of levels. [ABSTRACT FROM AUTHOR]

We study a classic problem introduced thirty years ago by Eades and Wormald. Let G = (V , E , λ) be a weighted planar graph, where λ : E → R + is a length function. The Fixed Edge-Length Planar Realization problem (FEPR for short) asks whether there exists a planar straight-line realization of G , i.e., a planar straight-line drawing of G where the Euclidean length of each edge e ∈ E is λ (e). Cabello, Demaine, and Rote showed that the FEPR problem is NP-hard, even when λ assigns the same value to all the edges and the graph is triconnected. Since the existence of large triconnected minors is crucial to the known NP-hardness proofs, in this paper we investigate the computational complexity of the FEPR problem for weighted 2-trees, which are K 4 -minor free. We show the NP-hardness of the problem, even when λ assigns to the edges only up to four distinct lengths. Conversely, we show that the FEPR problem is linear-time solvable when λ assigns to the edges up to two distinct lengths, or when the input has a prescribed embedding. Furthermore, we consider the FEPR problem for weighted maximal outerplanar graphs and prove it to be linear-time solvable if their dual tree is a path, and cubic-time solvable if their dual tree is a caterpillar. Finally, we prove that the FEPR problem for weighted 2-trees is slice-wise polynomial in the length of the large path. [ABSTRACT FROM AUTHOR]

In a drawing of a clustered graph vertices and edges are drawn as points and curves, respectively, while clusters are represented by simple closed regions. A drawing of a clustered graph is c-planar if it has no edge–edge, edge–region, or region–region crossings. Determining the complexity of testing whether a clustered graph admits a c-planar drawing is a long-standing open problem in the Graph Drawing research area. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, this condition is not sufficient and the consequences on the problem due to the requirement of not having edge–region and region–region crossings are not yet fully understood. In order to shed light on the c-planarity problem, we consider a relaxed version of it, where some kinds of crossings (either edge–edge, edge–region, or region–region) are allowed even if the underlying graph is planar. We investigate the relationships among the minimum number of edge–edge, edge–region, and region–region crossings for drawings of the same clustered graph. Also, we consider drawings in which only crossings of one kind are admitted. In this setting, we prove that drawings with only edge–edge or with only edge–region crossings always exist, while drawings with only region–region crossings may not. Further, we provide upper and lower bounds for the number of such crossings. Finally, we give a polynomial-time algorithm to test whether a drawing with only region–region crossings exists for biconnected graphs, hence identifying a first non-trivial necessary condition for c-planarity that can be tested in polynomial time for a noticeable class of graphs. [ABSTRACT FROM AUTHOR]