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We show that every plane graph admits a planar straight-line drawing in which all faces with more than three vertices are nonconvex polygons. [ABSTRACT FROM AUTHOR]

Consider an n-vertex planar graph G. The depth of an embedding Γ of G is the maximum distance of its internal faces from the external one. Several researchers pointed out that the quality of a planar embedding can be measured in terms of its depth. We present an O( n)-time algorithm for computing an embedding of G with minimum depth. This bound improves on the best previous bound by an O( nlog n) factor. As a side effect, our algorithm improves the bounds of several algorithms that require the computation of a minimum-depth embedding. [ABSTRACT FROM AUTHOR]

Graph Drawing is (usually) concerned with the production of readable representations of graphs. In this paper, instead of investigating how to produce “good” drawings we tackle the opposite problem of producing “bad” drawings. In particular, we study how to construct orthogonal drawings with many bends along the edges and with large area. Our results show surprising contact points, in Graph Drawing, between the computational cost of niceness and the one of ugliness. [ABSTRACT FROM AUTHOR]

We investigate the computational complexity of the following problem. Given a planar graph in which some vertices have already been placed in the plane, place the remaining vertices to form a planar straight-line drawing of the whole graph. We show that this extensibility problem, proposed in the 2003 "Selected Open Problems in Graph Drawing" [1], is NP-hard. [ABSTRACT FROM AUTHOR]

We propose a new metaphor for the visualization of prefixes propagation in the Internet. Such a metaphor is based on the concept of topographic map and allows to put in evidence the relative importance of the Internet Service Providers (lSPs) involved in the routing of the prefix. Based on the new metaphor we propose an algorithm for computing layouts and experiment with such algorithm on a test suite taken from the real Internet. The paper extends the visualization approach of the BGPIay service, which is an Internet routing monitoring tool widely used by ISP operators. [ABSTRACT FROM AUTHOR]

We introduce an open-source, scalable, and distributed architecture, called Megalos, that supports the implementation of virtual network scenarios consisting of virtual devices (VDs) where each VD may have several Layer 2 interfaces assigned to virtual LANs. We rely on Docker containers to realize vendor-independent VDs and we leverage Kubernetes for the management of the nodes of a distributed cluster. Our architecture does not require platform-specific configurations and supports a seamless interconnection between the virtual environment and the physical one. Also, it guarantees the segregation of each virtual LAN traffic from the traffic of other LANs, from the cluster traffic, and from Internet traffic. Further, a packet is only sent to the cluster node containing the recipient VD. We produce several example applications where we emulate large network scenarios, with thousands of VDs and LANs. Finally, we experimentally show the scalability potential of Megalos by measuring the overhead of the distributed environment and of its signaling protocols. [ABSTRACT FROM AUTHOR]

To reconstruct point geometry from multiple images, computation of the fundamental matrix is always necessary. With a new optimization criterion, i.e., the re-projective 3D metric geometric distance rather than projective space under RANSAC (Random Sample And Consensus) framework, our method can reveal the quality of the fundamental matrix visually through 3D reconstruction. The geometric distance is the projection error of 3D points to the corresponding image pixel coordinates in metric space. The reasonable visual figures of the reconstructed scenes are shown but only some numerical result were compared, as is standard practice. This criterion can lead to a better 3D reconstruction result especially in 3D metric space. Our experiments validate our new error criterion and the quality of fundamental matrix under the new criterion. [ABSTRACT FROM AUTHOR]

A k-page upward book embedding (kUBE) of a directed acyclic graph G is a book embeddings of G on k pages with the additional requirement that the vertices appear in a topological ordering along the spine of the book. The kUBE Testing problem, which asks whether a graph admits a kUBE, was introduced in 1999 by Heath, Pemmaraju, and Trenk (SIAM J Comput 28(4), 1999). In a companion paper, Heath and Pemmaraju (SIAM J Comput 28(5), 1999) proved that the problem is linear-time solvable for k = 1 and NP-complete for k = 6 . Closing this gap has been a central question in algorithmic graph theory since then. In this paper, we make a major contribution towards a definitive answer to the above question by showing that kUBE Testing is NP-complete for k ≥ 3 , even for st-graphs, i.e., acyclic directed graphs with a single source and a single sink. Indeed, our result, together with a recent work of Bekos et al. (Theor Comput Sci 946, 2023) that proves the NP-completeness of 2UBE for planar st-graphs, closes the question about the complexity of the kUBE problem for any k. Motivated by this hardness result, we then focus on the 2UBE Testing for planar st-graphs. On the algorithmic side, we present an O (f (β) · n + n 3) -time algorithm for 2UBE Testing, where β is the branchwidth of the input graph and f is a singly-exponential function on β . Since the treewidth and the branchwidth of a graph are within a constant factor from each other, this result immediately yields an FPT algorithm for st-graphs of bounded treewidth. Furthermore, we describe an O(n)-time algorithm to test whether a plane st-graph whose faces have a special structure admits a 2UBE that additionally preserves the plane embedding of the input st-graph. On the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2 UBE. [ABSTRACT FROM AUTHOR]

We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant k. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in O (k 3 k + 3 2 · n) time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in O(n) time for k = 2 , but it is NP-complete for any k > 2 . NodeTrix planarity testing remains NP-complete also in the free sides model when k > 4 . [ABSTRACT FROM AUTHOR]

Many different approaches have been proposed for the challenging problem of visually analyzing large networks. Clustering is one of the most promising. In this paper, we propose a new clustering technique whose goal is that of producing both intracluster graphs and intercluster graph with desired topological properties. We formalize this concept in the (X,Y) -clustering framework, where Y is the class that defines the desired topological properties of intracluster graphs and X is the class that defines the desired topological properties of the intercluster graph. By exploiting this approach, hybrid visualization tools can effectively combine different node-link and matrix-based representations, allowing users to interactively explore the graph by xpansion/contraction of clusters without loosing their mental map. As a proof of concept, we describe the system Visual Hybrid (X,Y)-clustering (VHYXY) that implements our approach and we present the results of case studies to the visual analysis of social networks. [ABSTRACT FROM PUBLISHER]

Abstract: Mixed graphs have both directed and undirected edges and have received considerable attention in the literature. We study two upward planarity testing problems for embedded mixed graphs, give some NP-hardness results, and describe Integer Linear Programming techniques to solve them. Experiments show the efficiency of our approach. [Copyright &y& Elsevier]

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 this paper we study planar morphs between planar straight-line grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees. [ABSTRACT FROM AUTHOR]

We prove that, given two topologically-equivalent upward planar straight-line drawings of an n-vertex directed graph G, there always exists a morph between them such that all the intermediate drawings of the morph are upward planar and straight-line. Such a morph consists of O(1) morphing steps if G is a reduced planar st-graph, O(n) morphing steps if G is a planar st-graph, O(n) morphing steps if G is a reduced upward planar graph, and O (n 2) morphing steps if G is a general upward planar graph. Further, we show that Ω (n) morphing steps might be necessary for an upward planar morph between two topologically-equivalent upward planar straight-line drawings of an n-vertex path. [ABSTRACT FROM AUTHOR]