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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]

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems. [ABSTRACT FROM AUTHOR]

An ortho-polygon visibility representation of an n-vertex embedded graph G (OPVR of G) is an embedding-preserving drawing of G that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of G is the minimum k such that every polygon has at most k reflex corners. We present polynomial time algorithms that test whether G has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of G are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if G has at most one crossing per edge (i.e., G is a 1-plane graph), an OPVR of G always exists while this may not be the case if two crossings per edge are allowed. Also, if G is a 3-connected 1-plane graph, we can compute an OPVR of G whose vertex complexity is bounded by a constant in O(n) time. However, if G is a 2-connected 1-plane graph, the vertex complexity of any OPVR of G may be Ω(n). In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in O(n) time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs. [ABSTRACT FROM AUTHOR]

A drawing of a graph is 1-planar if each edge is crossed at most once. A graph is 1-planar if it has a 1-planar drawing. A k-bend RAC (Right Angle Crossing) drawing of a graph is a polyline drawing where each edge has at most k bends and edges cross only at right angles. A graph is k-bend RAC if it has a k -bend RAC drawing. A 0-bend RAC graph (drawing) is also called a straight-line RAC graph (drawing) . The relationships between 1-planar and k -bend RAC graphs have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC and straight-line RAC graphs that are not 1-planar. The existence of 1-planar straight-line RAC drawings has been proven only for restricted families of 1-planar graphs. Two of the main questions still open are: ( i ) What is the complexity of deciding whether a graph has a drawing that is both 1-planar and straight-line RAC? ( i i ) Does every 1-planar graph have a drawing that is both 1-planar and 1-bend RAC? In this paper we answer these two questions. Namely, we prove an NP-hardness result for the first question, and we positively answer the second question by describing a drawing algorithm for 1-planar graphs. [ABSTRACT FROM AUTHOR]

A mixed graph has both directed and undirected edges. We study how to compute a crossing-free drawing of an embedded planar mixed graph, such that it is upward 'as much as possible'. Roughly speaking, in an upward drawing of a mixed graph all (undirected) edges are monotone in the vertical direction and directed edges flow monotonically from bottom to top according to their orientation. We study quasi-upward drawings of mixed graphs, that is, upward drawings where edges can break the vertical monotonicity in a finite number of edge points, called bends. We describe both efficient heuristic techniques and exact approaches for computing quasi-upward planar drawings of embedded mixed graphs with few bends, and we extensively compare them experimentally: the results suggest that our algorithms are effective in many cases. [ABSTRACT FROM AUTHOR]

Abstract: We propose an approach that allows a user (e.g., an analyst) to explore a layout produced by any graph drawing algorithm, in order to reduce the visual complexity and clarify its presentation. Our approach is based on stratifying the drawing into layers with desired properties; to this aim, heuristics are presented. The produced layers can be explored and combined by the user to gradually acquire details. We present a user study to test the effectiveness of our approach. Furthermore, we performed an experimental analysis on popular force-directed graph drawing algorithms, in order to evaluate what is the algorithm that produces the smallest number of layers and if there is any correlation between the number of crossings and the number of layers of a graph layout. The proposed approach is useful to explore graph layouts, as confirmed by the presented user study. Furthermore, interesting considerations arise from the experimental evaluation, in particular, our results suggest that the number of layers of a graph layout may represent a reliable measure of its visual complexity. The algorithms presented in this paper can be effectively applied to graph layouts with a few hundreds of edges and vertices. For larger drawings that contain lots of crossings, the time complexity of our algorithms grows quadratically in the number of edges and more efficient techniques need to be devised. The proposed approach takes as input a layout produced by any graph drawing algorithm, therefore it can be applied in a variety of application domains. Several research directions can be explored to extend our framework and to devise new visualization paradigms to effectively present stratified drawings. [Copyright &y& Elsevier]

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]

Abstract: In this paper we study how to compute compact straight-line drawings of planar graphs with a limited number of crossings per edge. We prove that every outerplanar graph can be drawn in area using a sub-linear number of crossings per edge, and that for any given number , every outerplanar graph admits an area drawing with crossing per edge. The drawing algorithms run in linear time and can be also generalized to another meaningful sub-family of series–parallel graphs with bounded vertex-degree. [Copyright &y& Elsevier]

Abstract: The crossing resolution of a geometric graph is the minimum crossing angle at which any two edges cross each other. In this paper, we present upper and lower bounds to the crossing resolution of the complete geometric graphs. [Copyright &y& Elsevier]

Abstract: Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing. This paper initiates the study of combinatorial and algorithmic questions related to the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed. [Copyright &y& Elsevier]

A digraph is upward planar if it admits a planar drawing where all edges are monotone in the upward direction. It is known that the problem of testing a digraph for upward planarity is NP-complete in general. This paper describes an O(n4)-time upward planarity testing algorithm for all digraphs that have a series-parallel structure, where n is the number of vertices of the input. This significantly enlarges the family of digraphs for which a polynomial-time testing algorithm is known. Furthermore, the study is extended to general digraphs, and a fixed parameter tractable algorithm for upward planarity testing is described, whose time complexity is O(dt · t · n3 + d · t2 · n + d2 · n2) where t is the number of triconnected components of the digraph and d is an upper bound on the diameter of any split component of the digraph. Our results use the new notion of upward spirality that, informally speaking, is a measure of the "level of winding" that a triconnected component of a digraph G can have in an upward planar drawing of G. [ABSTRACT FROM AUTHOR]

Abstract: Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of . We concentrate on trees and show how to compute the output in time in a real-RAM model and with at most edges with at most bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least bends on edges. [Copyright &y& Elsevier]

Abstract: Let G be an embedded planar digraph. A maximum upward planar subgraph of G is an embedding preserving subgraph that is upward planar and, among those, has the maximum number of edges. This paper presents an extensive study on the problem of computing maximum upward planar subgraphs of embedded planar digraphs: Complexity results, algorithms, and experiments are presented. Namely: (i) we prove that the addressed problem is NP-Hard; (ii) a fast heuristic and an exponential-time exact algorithm are described; (iii) a wide experimental analysis is performed to show the effectiveness of our techniques. [Copyright &y& Elsevier]

Abstract: In this paper, we study k-spine, h-bend planar drawings in which each vertex of a planar graph G lies on one of horizontal lines and each edge of G is drawn as a polyline containing at most bends. A graph with a k-spine, h-bend planar drawing is said to be k-spine, h-bend planar. We mainly focus on k-spine, 1-bend planar drawings, showing that for each , there exists a planar graph that is not k-spine, 1-bend planar, and furthermore, that it is -hard to test k-spine, 1-bend planarity. Given this complexity result, we further narrow our focus onto 2-spine, 1-bend planar drawings. We characterize 2-spine, 1-bend planarity using a new generalization of Hamiltonian graphs that we call Hamiltonian-with-handles graphs. We observe that our characterization naturally extends the connection between 2-page book embeddings and Hamiltonicity. Finally, we use our characterization to show that 2-outerplanar graphs are 2-spine, 1-bend planar. [Copyright &y& Elsevier]