Search

Given a graph G and an integer b , OrthogonalPlanarity is the problem of testing whether G admits a planar orthogonal drawing with at most b bends. OrthogonalPlanarity is known to be NP -complete. We show that this problem belongs to the XP class when parameterized by treewidth. The proof exploits a fixed-parameter tractable approach that uses two more parameters besides treewidth, namely the natural parameter b and the number of vertices with degree at most two of G. Such approach is based on the new concept of sketched orthogonal representation, which synthetically describes a family of equivalent orthogonal drawings. The approach is general enough to be applicable to other related problems, namely HV-Planarity and FlexDraw. Also, in the special case of series-parallel graphs we obtain that both OrthogonalPlanarity and HV-Planarity can be solved in O (n 3 log ⁡ n) time, which improves on the previous O (n 4) bounds for these two. [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]

Abstract An HV-graph is a planar graph with vertex-degree at most four such that each edge is labeled either H (horizontal) or V (vertical). The HV-planarity testing problem asks whether an HV-graph admits an HV-drawing , that is, a planar drawing such that each edge with label H is drawn as a horizontal segment and each edge with label V is drawn as a vertical segment. We prove that the HV-planarity testing problem is NP-complete even for graphs with vertex-degree at most three, which answers an open question posed by both Manuch et al. [30] and Durocher et al. [17]. On the positive side, we prove that the HV-planarity testing problem can be solved in polynomial-time for series-parallel graphs. This result significantly extends a previous result by Durocher et al. about HV-planarity testing of biconnected outerplanar graphs of maximum vertex-degree three. Highlights • HV-planarity testing is NP-complete, even for graphs with vertex-degree at most 3. • Polynomial-time HV-planarity testing for 2-connected series-parallel graphs. • Polynomial-time HV-planarity testing for partial 2-trees. [ABSTRACT FROM AUTHOR]

We study the problem of representing topological graphs as polyline drawings with few bends per edge and such that the topology of the graph is either fully or partially preserved. More formally, let G be a simple topological graph and let Γ be a polyline drawing of G. Drawing Γ partially preserves the topology of G if it has the same external boundary, the same circular order of the edges around each vertex, and the same set of crossings as G , while it fully preserves the topology of G if the planarization of G and the planarization of Γ have the same planar embedding. We prove that if the set of crossing-free edges of G forms a biconnected (connected) spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most one (three), i.e., with at most one (three) bend(s) per edge. If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Ω (n) , while it is O (1) if the number of connected components is O (1). Concerning drawings that fully preserve the topology, we show that if G is k -skew (i.e., it becomes planar after removing k suitably chosen edges), it admits one such drawing with curve complexity at most 2 k ; for 1-skew graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs (i.e., with at most two crossings per edge and maximum edge density), for which we discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology. [ABSTRACT FROM AUTHOR]

Abstract We show that any outerplanar graph admits a planar straight-line drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any ϵ > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 − ϵ. We also show that this ratio cannot be bounded if the embeddings of the outerplanar graphs are given. [ABSTRACT FROM AUTHOR]

A simple topological graph is k -quasiplanar (k ≥ 2) if it contains no k pairwise crossing edges, and k -planar if no edge is crossed more than k times. In this paper, we explore the relationship between k -planarity and k -quasiplanarity to show that, for k ≥ 2 , every k -planar simple topological graph can be transformed into a (k + 1) -quasiplanar simple topological graph. [ABSTRACT FROM AUTHOR]

Visibility representations are a well-known paradigm to represent graphs. From a high-level perspective, a visibility representation of a graph G maps the vertices of G to non-overlapping geometric objects and the edges of G to visibilities, i.e., segments that do not intersect any geometric object other than at their end-points. In this paper, we initiate the study of 2.5D box visibility representations (2.5D-BRs) where vertices are mapped to 3D boxes having the bottom face in the plane z = 0 and edges are unobstructed lines of sight parallel to the x - or y -axis. We prove that: ( i ) Every complete bipartite graph admits a 2.5D-BR; ( i i ) The complete graph K n admits a 2.5D-BR if and only if n ⩽ 19 ; ( i i i ) Every graph with pathwidth at most 7 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBRs), which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an n -vertex graph that admits a 2.5D-GBR has at most 4 n − 6 n edges and this bound is tight. Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete (the footprint of a 2.5D-BR Γ is the set of bottom faces of the boxes in Γ). [ABSTRACT FROM AUTHOR]

A 1 -plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a 1-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a 1-plane graph G is a coloring of the edges of G with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: ( i ) Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. ( ii ) Deciding whether a NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. ( iii ) Deciding whether a 1-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed. [ABSTRACT FROM AUTHOR]

We give new results about the relationship between 1-planar graphs and RAC graphs . A graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is RAC if it can be drawn in such a way that its edges cross only at right angles. These two classes of graphs and their relationships have been widely investigated in the last years, due to their relevance in application domains where computing readable graph layouts is important to analyze or design relational data sets. We study IC-planar graphs , the sub-family of 1-planar graphs that admit 1-planar drawings with independent crossings (i.e., no two crossed edges share an endpoint). We prove that every IC-planar graph admits a straight-line RAC drawing, which may require however exponential area. If we do not require right angle crossings, we can draw every IC-planar graph with straight-line edges in linear time and quadratic area. We then study the problem of testing whether a graph is IC-planar. We prove that this problem is NP-hard, even if a rotation system for the graph is fixed. On the positive side, we describe a polynomial-time algorithm that tests whether a triangulated plane graph augmented with a given set of edges that form a matching is IC-planar. [ABSTRACT FROM AUTHOR]