We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.

We study the following question, which has been considered since the 90’s: Does every st-planar graph admit a planar straight-line dominance drawing? We show concrete evidence for the difficulty of this question, by proving that, unlike upward planar straight-line drawings, planar straight-line dominance drawings with prescribed y-coordinates do not always exist and planar straight-line dominance drawings cannot always be constructed via a contract-draw-expand inductive approach. We also show several classes of st-planar graphs that always admit a planar straight-line dominance drawing. These include st-planar 3-trees in which every stacking operation introduces two edges incoming into the new vertex, st-planar graphs in which every vertex is adjacent to the sink, and st-planar graphs in which no face has the left boundary that is a single edge.

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward $k$-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most $k$ times for some integer $k \ge 1$. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that testing upward-$k$-planarity is NP-complete already for $k=1$ and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

Abstract A map is a partition of the sphere into interior-disjoint regions homeomorphic to closed disks. Some regions are labeled as nations, while the remaining ones are labeled as holes. A map in which at mostknations touch at the same point is ak-map, while it is hole-free if it contains no holes. A graph is a map graph if there is a bijection between its vertices and the nations of a map, such that two nations touch if and only the corresponding vertices are connected by an edge. We present a fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. Its time complexity is linear in the size of the graph. It reports a certificate in the form of a so-called witness, if the input is a yes-instance. Our algorithmic framework is general enough to test, for anyk, if the input graph admits ak-map or a hole-free k-map.

Abstract The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs where vertices of the two parts lie on two horizontal lines and edges lie between these lines. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been considered only for $k=1$ in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of $2$-layer $k$-planar graphs with $k\in \{2,3,4,5\}$. Based on these results, we provide a Crossing Lemma for $2$-layer $k$-planar graphs, which then implies a general density bound for $2$-layer $k$-planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between $k$-planarity and $h$-quasiplanarity in the $2$-layer model and show that $2$-layer $k$-planar graphs have pathwidth at most $k+1$ while there are also $2$-layer $k$-planar graphs with pathwidth at least $(k+3)/2$.

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model.

Abstract Bitonic st-orderings for st-planar graphs were introduced as a method to cope with several graph drawing problems. Notably, they have been used to obtain the best-known upper bound on the number of bends for upward planar polyline drawings with at most one bend per edge in polynomial area. For an st-planar graph that does not admit a bitonic st-ordering, one may split certain edges such that for the resulting graph such an ordering exists. Since each split is interpreted as a bend, one is usually interested in splitting as few edges as possible. While this optimization problem admits a linear-time algorithm in the fixed embedding setting, it remains open in the variable embedding setting. We close this gap in the literature by providing a linear-time algorithm that optimizes over all embeddings of the input st-planar graph. The best-known lower bound on the number of required splits of an st-planar graph with n vertices is $$n-3$$ n - 3 . However, it is possible to compute a bitonic st-ordering without any split for the st-planar graph obtained by reversing the orientation of all edges. In terms of upward planar polyline drawings in polynomial area, the former translates into $$n-3$$ n - 3 bends, while the latter into no bends. We show that this idea cannot always be exploited by describing an st-planar graph that needs at least $$n-5$$ n - 5 splits in both orientations. We provide analogous bounds for graphs with small degree. Finally, we further investigate the relationship between splits in bitonic st-orderings and bends in upward planar polyline drawings with polynomial area, by providing bounds on the number of bends in such drawings.

Abstract A morph is a continuous transformation between two representations of a graph. We consider the problem of morphing between contact representations of a plane graph. In an $${\mathcal {F}}$$ F -contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family $${\mathcal {F}}$$ F of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in G. In a morph between two $${\mathcal {F}}$$ F -contact representations we insist that at each time step (continuously throughout the morph) we have an $${\mathcal {F}}$$ F -contact representation. We focus on the case when $$\mathcal {F}$$ F is the family of triangles in $$\mathbb {R}^2$$ R 2 that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Moreover, they naturally correspond to 3-orientations. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We characterize the pairs of RT-representations admitting a morph between each other via the respective 3-orientations. Our characterization leads to a polynomial-time algorithm to decide whether there is a morph between two RT-representations of an n-vertex plane triangulation, and, if so, computes a morph with $${\mathcal {O}}(n^2)$$ O ( n 2 ) steps. Each of these steps is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. Our characterization also implies that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the “top-most” triangle in both representations corresponds to the same vertex.

Motivated by cognitive experiments providing evidence that large crossing-angles do not impair the readability of a graph drawing, RAC (Right Angle Crossing) drawings were introduced to address the problem of producing readable representations of non-planar graphs by supporting the optimal case in which all crossings form 90° angles. In this work, we make progress on the problem of finding RAC drawings of graphs of low degree. In this context, a long-standing open question asks whether all degree-3 graphs admit straight-line RAC drawings. This question has been positively answered for the Hamiltonian degree-3 graphs. We improve on this result by extending to the class of 3-edge-colorable degree-3 graphs. When each edge is allowed to have one bend, we prove that degree-4 graphs admit such RAC drawings, a result which was previously known only for degree-3 graphs. Finally, we show that 7-edge-colorable degree-7 graphs admit RAC drawings with two bends per edge. This improves over the previous result on degree-6 graphs.

In an upward-planar L-drawing of a directed acyclic graph (DAG) each edge $e$ is represented as a polyline composed of a vertical segment with its lowest endpoint at the tail of $e$ and of a horizontal segment ending at the head of $e$. Distinct edges may overlap, but not cross. Recently, upward-planar L-drawings have been studied for $st$-graphs, i.e., planar DAGs with a single source $s$ and a single sink $t$ containing an edge directed from $s$ to $t$. It is known that a plane $st$-graph, i.e., an embedded $st$-graph in which the edge $(s,t)$ is incident to the outer face, admits an upward-planar L-drawing if and only if it admits a bitonic $st$-ordering, which can be tested in linear time. We study upward-planar L-drawings of DAGs that are not necessarily $st$-graphs. On the combinatorial side, we show that a plane DAG admits an upward-planar L-drawing if and only if it is a subgraph of a plane $st$-graph admitting a bitonic $st$-ordering. This allows us to show that not every tree with a fixed bimodal embedding admits an upward-planar L-drawing. Moreover, we prove that any acyclic cactus with a single source (or a single sink) admits an upward-planar L-drawing, which respects a given outerplanar embedding if there are no transitive edges. On the algorithmic side, we consider DAGs with a single source (or a single sink). We give linear-time testing algorithms for these DAGs in two cases: (i) when the drawing must respect a prescribed embedding and (ii) when no restriction is given on the embedding, but it is biconnected and series-parallel.

Given a bipartite graph G = (Vb, Vr, E), the 2-Level Quasi-Planarity problem asks for the existence of a drawing of G in the plane such that the vertices in Vb and in Vr lie along two parallel lines ℓb and ℓr, respectively, each edge in E is drawn in the unbounded strip of the plane delimited by ℓb and ℓr, and no three edges in E pairwise cross. We prove that the 2-LEVEL Quasi-Planarity problem is NP-complete. This answers an open question of Dujmović, Pór, and Wood. Furthermore, we show that the problem becomes linear-time solvable if the ordering of the vertices in Vb along ℓb is prescribed. Our contributions provide the first results on the computational complexity of recognizing quasi-planar graphs, which is a long-standing open question. Our linear-time algorithm exploits several ingredients, including a combinatorial characterization of the positive instances of the problem in terms of the existence of a planar embedding with a caterpillar-like structure, and an SPQR-tree-based algorithm for testing the existence of such a planar embedding. Our algorithm builds upon a classification of the types of embeddings with respect to the structure of the portion of the caterpillar they contain and performs a computation of the realizable embedding types based on a succinct description of their features by means of constant-size gadgets.

We study the planar orthogonal drawing style within the framework of partial representation extension. Let $$(G,H,\varGamma _H)$$ be a partial orthogonal drawing, i.e., G is a graph, $$H\subseteq G$$ is a subgraph and $$\varGamma _H$$ is a planar orthogonal drawing of H. We show that the existence of an orthogonal drawing $$\varGamma _G$$ of G that extends $$\varGamma _H$$ can be tested in linear time. If such a drawing exists, then there also is one that uses O(|V(H)|) bends per edge. On the other hand, we show that it is NP-complete to find an extension that minimizes the number of bends or has a fixed number of bends per edge.

We consider the problem of morphing between contact representations of a plane graph. In a contact representation of a plane graph, vertices are realized by internally disjoint elements from a family of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in the graph. In a morph between two contact representations we insist that at each time step (continuously throughout the morph) we have a contact representation of the same type. We focus on the case when the geometric objects are triangles that are the lower-right half of axis-parallel rectangles. Such RT-representations exist for every plane graph and right triangles are one of the simplest families of shapes supporting this property. Thus, they provide a natural case to study regarding morphs of contact representations of plane graphs. We study piecewise linear morphs, where each step is a linear morph moving the endpoints of each triangle at constant speed along straight-line trajectories. We provide a polynomial-time algorithm that decides whether there is a piecewise linear morph between two RT-representations of a plane triangulation, and, if so, computes a morph with a quadratic number of linear morphs. As a direct consequence, we obtain that for 4-connected plane triangulations there is a morph between every pair of RT-representations where the "top-most" triangle in both representations corresponds to the same vertex. This shows that the realization space of such RT-representations of any 4-connected plane triangulation forms a connected set.

Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size of the largest complete or complete bipartite graph belonging to it. Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and complete bipartite graphs, taking into account the constraints of the specific class. As a proof of concept, we apply our technique to various beyond-planarity classes and achieve new tight bounds for the aforementioned parameter.

We study the version of the C-Planarity problem in which edges connecting the same pair of clusters must be grouped into pipes, which generalizes the Strip Planarity problem. We give algorithms to decide several families of instances for the two variants in which the order of the pipes around each cluster is given as part of the input or can be chosen by the algorithm.

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 W indrose P lanarity 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 2 n −5 bends in total, which lies on the 3 n × 3 n grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).

Given an $n$-vertex graph and two straight-line planar drawings of the graph that have the same faces and the same outer face, we show that there is a morph (i.e., a continuous transformation) between the two drawings that preserves straight-line planarity and consists of $O(n)$ steps, which we prove is optimal in the worst case. Each step is a unidirectional linear morph, which means that every vertex moves at constant speed along a straight line, and the lines are parallel although the vertex speeds may differ. Thus we provide an efficient version of Cairns' 1944 proof of the existence of straight-line planarity-preserving morphs for triangulated graphs, which required an exponential number of steps.

Given a planar graph G(V, E) and a partition of the neighbors of each vertex v ∊ V in four sets , and , 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 ∊ is above and to the right of v, (ii) each neighbor u ∊ is above and to the left of v, (iii) each neighbor u ∊ is below and to the left of v, (iv) each neighbor u ∊ 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 to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is -hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a combinatorial embedding that is given as part of the input. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph admitting a windrose-planar drawing we show how to construct one with at most one bend per edge on an O(n) × O(n) grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.

In this article, we deepen the understanding of the connection between two long-standing graph drawing open problems, Simultaneous Embedding with Fixed Edges (SEFE-2) and Clustered Planarity (C-Planarity). Given two planar graphs on the same set of vertices, the SEFE-2 problem asks to find planar drawings of the two graphs such that each vertex lies on the same point and each common edge is represented by the same curve in both drawings. Given a planar graph together with a recursive clustering of its vertices, the C-Planarity problem asks to find a planar drawing of the graph and a representation of each cluster as a simple region enclosing all and only the vertices of the cluster such that no unnecessary intersection involving clusters and edges is created. In a recent article at GD’12, Marcus Schaefer presented a reduction from C-Planarity to SEFE-2. We prove that a reduction exists also in the opposite direction, if we restrict to instances of SEFE-2 in which the graph induced by the common edges is connected. We pose as an intriguing open question whether the two problems are polynomial-time equivalent.

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.

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.

In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n4) steps.

Abstract A greedy drawing is a graph drawing containing a distance‐decreasing path for every pair of nodes. A path ( v 0 , v 1 ,…, v m ) is distance‐decreasing if d ( v i , v m ) < d ( v i ‐1 , v m ), for i = 1,…, m . Greedy drawings easily support geographic greedy routing. Hence, a natural and practical problem is the one of constructing greedy drawings in the plane using few bits for representing vertex Cartesian coordinates and using the Euclidean distance as a metric. We show that there exist greedy‐drawable graphs that do not admit any greedy drawing in which the Cartesian coordinates have less than a polynomial number of bits. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

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 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 hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “on-cas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) 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 improve our algorithm to reach linear-time complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.