Search

This paper studies a packing problem in the so-called beyond-planar setting, that is when the host graph is "almost-planar" in some sense. Precisely, we consider the case that the host graph is k -planar, i.e., it admits an embedding with at most k crossings per edge, and focus on families of Δ -regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree Δ. We study the dependency of k from the number h of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called placement) and when they are not. We give necessary and sufficient conditions for the placement of h Δ -regular caterpillars and sufficient conditions for the packing of a set of Δ 1 -, Δ 2 -, ... , Δ h -regular caterpillars such that the degree Δ i and the degree Δ j of the non-leaf vertices can differ from one caterpillar to another, for 1 ≤ i , j ≤ h , i ≠ j. [ABSTRACT FROM AUTHOR]

We study universal sets of slopes for computing upward planar drawings of planar st-graphs. We first consider a subfamily of planar st-graphs, called bitonic st-graphs. We prove that every set S of Δ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree Δ , i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in S . This result is worst-case optimal in terms of number of slopes, and, for a suitable choice of S , it gives rise to drawings with worst-case optimal angular resolution. We then prove that every such set S can be used to construct 2-bend upward planar drawings of n-vertex planar st-graphs with at most 4 n - 9 bends in total. [ABSTRACT FROM AUTHOR]

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]

We study the problem of representing a graph as a storyplan, a recently introduced model for dynamic graph visualization. It is based on a sequence of frames, each showing a subset of vertices and a planar drawing of their induced subgraphs, where vertices appear and disappear over time. Namely, in the StoryPlan problem, we are given a graph and we want to decide whether there exists a total vertex appearance order for which a storyplan exists. We prove that the problem is NP -complete, and complement this hardness with two parameterized algorithms, one in the vertex cover number and one in the feedback edge set number of the input graph. We prove that partial 3-trees always admit a storyplan, which can be computed in linear time. Finally, we show that the problem remains NP -complete if the vertex appearance order is given and we have to choose how to draw the frames. [ABSTRACT FROM AUTHOR]

• For n ≥ 14 there is a properly 2-colored SP-graph with 2 n + 4 vertices s.t. any 2-colored PSE requires Ω (n) bends on Ω (n) edges • 2 bends per edge suffice for a 2-colored PSE of properly 2-colored outerplanar graphs • 1 bend per edge suffices for a 2-colored PSE of a properly 2-colored outerplanar graph if the points are linearly separable • 3 bends per edge suffice for a 2-colored PSE of a 2-colored outerplanar graph • 1 bend per edge suffices for a 2-colored PSE of a 2-colored caterpillar Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (resp. blue) points as the red (resp. blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (resp. blue) vertex of G to a red (resp. blue) point of S. We show that there exist partial 2-trees that are properly 2-colored (i.e., they are 2-colored with no two adjacent vertices have the same color), whose point-set embeddings may require linearly many bends on linearly many edges. For a contrast, we show that two bends per edge are sufficient for 2-colored point-set embedding of properly 2-colored outerplanar graphs. For separable point sets this bound reduces to one, which is worst-case optimal. If the 2-coloring of the outerplanar graph is not proper, three bends per edge are sufficient and one bend per edge (which is worst-case optimal) is sufficient for caterpillars. [ABSTRACT FROM AUTHOR]

We present a new model for hybrid planarity that relaxes existing hybrid representation models. A graph G = (V , E) is (k , p) -planar if V can be partitioned into clusters of size at most k such that G admits a drawing where: (i) each cluster is associated with a closed, bounded planar region, called a cluster region ; (ii) cluster regions are pairwise disjoint, (iii) each vertex v ∈ V is identified with at most p distinct points, called ports , on the boundary of its cluster region; (iv) each inter-cluster edge (u , v) ∈ E is identified with a Jordan arc connecting a port of u to a port of v ; (v) inter-cluster edges do not cross or intersect cluster regions except at their end-points. We first tightly bound the number of edges in a (k , p) -planar graph with p < k. We then prove that (4 , 1) -planarity testing and (2 , 2) -planarity testing are NP-complete problems. Finally, we prove that neither the class of (2 , 2) -planar graphs nor the class of 1-planar graphs contains the other, indicating that the (k , p) -planar graphs are a large and novel class. [ABSTRACT FROM AUTHOR]

We establish new results on the curve complexity of k -colored point-set embeddings when k = 3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require Ω (n 1 3 ) bends on Ω (n 2 3 ) edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k -colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has O (1) independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k -colored simultaneous embeddability problem and study its relationship with the k -colored point-set embeddability problem. [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]

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]

Abstract A topological graph is a graph drawn in the plane. A topological graph is k -plane, k > 0 , if each edge is crossed at most k times. We study the problem of partitioning the edges of a k -plane graph such that each partite set forms a graph with a simpler structure. While this problem has been studied for k = 1 , we focus on optimal 2-plane and on optimal 3-plane graphs, which are 2-plane and 3-plane graphs with maximum density. We prove the following results. (i) It is not possible to partition the edges of a simple (i.e., with neither self-loops nor parallel edges) optimal 2-plane graph into a 1-plane graph and a forest, while (ii) an edge partition formed by a 1-plane graph and two plane forests always exists and can be computed in linear time. (iii) There exist efficient algorithms to partition the edges of a simple optimal 2-plane graph into a 1-plane graph and a plane graph with maximum vertex degree at most 12, or with maximum vertex degree at most 8 if the optimal2-plane graph is such that its crossing-free edges form a graph with no separating triangles. (iv) There exists an infinite family of simple optimal 2-plane graphs such that in any edge partition composed of a 1-plane graph and a plane graph, the plane graph has maximum vertex degree at least 6 and the 1-plane graph has maximum vertex degree at least 12. (v) Every optimal 3-plane graph whose crossing-free edges form a biconnected graph can be decomposed, in linear time, into a 2-plane graph and two plane forests. [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 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]

A subcubic planar graph is a planar graph whose vertices have degree at most 3. We show that the subcubic planar graphs with at least five vertices have planar slope number at most 4, which is worst-case optimal. This answers an open question by Jelínek et al. [10] . Furthermore, we prove that the subcubic planar graphs with at least five vertices have angular resolution π / 4 , which solves an open problem by Kant [11] and by Formann et al. [8] . [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 study the problem of computing drawings of planar graphs in sub-quadratic area, by allowing edge crossings. We first prove that sub-quadratic area cannot be achieved if only a constant number of crossings per edge is allowed. More precisely, we show that the same area lower bounds as in the crossing-free case hold for straight-line and poly-line drawings of planar graphs and seriesparallel graphs. Motivated by this result, we study straight-line drawings of planar graphs where the number of crossings per edge is not bounded by a constant. In this case, we prove that every planar graph admits a straight-line drawing with sub-quadratic area and sub-linear thickness (the thickness of a drawing is the minimum number of colors that can be assigned to the edges so that each color class induces a planar drawing). We also prove that every partial 2-tree (and hence every series-parallel graph) admits a linear-area straight-line drawing with thickness at most 10. It is worth remarking that a drawing with thickness h - 1 is h-quasi planar, i.e. it does not contain h-mutually crossing edges. The main ingredient to prove our results is (c, t)-track layouts, a combinatorial tool that can be represented as a drawing where: (i) each vertex is assigned to one of t horizontal layers (tracks), (ii) no two adjacent vertices are on the same track, (iii) each edge receives one of c colors, so that no two edges of the same color (u, v) and (w, z) cross if u, w are on the same track, and v, z are on the same track. [ABSTRACT FROM AUTHOR]

Let G be a 3-connected planar graph with n vertices and let p ( G ) be the maximum number of vertices of an induced subgraph of G that is a path. Substantially improving previous results, we prove that p ( G ) ≥ log ⁡ n 12 log ⁡ log ⁡ n . To demonstrate the tightness of this bound, we notice that the above inequality implies p ( G ) ∈ Ω ( ( log 2 ⁡ n ) 1 − ε ) , where ε is any positive constant smaller than 1, and describe an infinite family of planar graphs for which p ( G ) ∈ O ( log ⁡ n ) . As a byproduct of our research, we prove a result of independent interest: Every 3-connected planar graph with n vertices contains an induced subgraph that is outerplanar and connected and that contains at least n 3 vertices. The proofs in the paper are constructive and give rise to O ( n ) -time algorithms. [ABSTRACT FROM AUTHOR]

A simultaneous geometric embedding (SGE) of two planar graphs G1 and G2 with the same vertex set is a pair of straight-line planar drawings Γ1 of G1 and Γ2 of G2 such that each vertex is drawn at the same point in Γ1 Γ2. Many papers have been devoted to the study of which pairs of graphs admit a SGE, and both positive and negative results have been proved. We extend the study of SGE, by introducing and characterizing a new class of planar graphs that makes it possible to immediately extend several positive results that rely on the property of strictly monotone paths. Moreover, we introduce a relaxation of the SGE setting where Γ1 and Γ2 are required to be quasi-planar (i.e. they can have crossings provided that there are no three mutually crossing edges). This relaxation allows for the simultaneous embedding of pairs of planar graphs that are not simultaneously embeddable in the classical SGE setting and opens up several new interesting research questions. [ABSTRACT FROM AUTHOR]

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. Fan-planar drawings have been recently introduced by Kaufmann and Ueckerdt [35] , who proved that every n -vertex fan-planar drawing has at most 5 n − 10 edges, and that this bound is tight for n ≥ 20 . We extend their result from both the combinatorial and the algorithmic point of view. We prove tight bounds on the density of constrained versions of fan-planar drawings and study the relationship between fan-planarity and k -planarity. Also, we prove that testing fan-planarity in the variable embedding setting is NP-complete. [ABSTRACT FROM AUTHOR]

Let ε, ε be two non negative numbers. An approximate rectangle of influence drawing (also called ( ε, ε)-RID) is a proximity drawing of a graph such that: (i) if u, v are adjacent vertices then their rectangle of influence 'scaled down' by the factor $\frac{1}{1+\varepsilon_{1}}$ does not contain other vertices of the drawing; (ii) if u, v are not adjacent, then their rectangle of influence 'blown-up' by the factor 1+ ε contains some vertices of the drawing other than u and v. Firstly, we prove that all planar graphs have an ( ε, ε)-RID for any ε>0 and ε>0, while there exist planar graphs which do not admit an ( ε,0)-RID and planar graphs which do not admit a (0, ε)-RID. Then, we investigate (0, ε)-RIDs; we prove that both the outerplanar graphs and a suitably defined family of graphs without separating 3-cycles admit this type of drawing. Finally, we study polynomial area approximate rectangle of influence drawings and prove that (0, ε)-RIDs of proper track planar graphs (a superclass of the outerplanar graphs) can be computed in polynomial area for any ε>2. As for values of ε such that ε≤2, we describe a drawing algorithm that computes (0, ε)-RIDs of binary trees in area $O(n^{c + f(\varepsilon_{2})})$, where c is a positive constant, f( ε) is a poly-logarithmic function that tends to infinity as ε tends to zero, and n is the number of vertices of the input tree. [ABSTRACT FROM AUTHOR]

A2-layer drawing of a bipartite graph G is a drawing such that the vertices of each partition set are drawn as points of a distinct horizontal line (called a layer) and the edges are drawn as straight-line segments. We study 2-layer drawings where edges can cross only at right angles; these drawings are called 2-layer right angle crossing drawings (2-layer RAC drawings for short). We focus on the following problem, which we call the maximum 2-layer RAC subgraph (M2LRacS) problem. Given a bipartite graph G, compute a subgraph H of G such that: (i) H admits a 2-layer RAC drawing and (ii) H has the maximum number of edges among the subgraphs of G that satisfy (i). We study this problem both in the no-fixed-layer setting, where no restriction is given on the vertex ordering on each layer, and in the 1-?xed-layer setting, where the ordering of the vertices of one of the two layers is given as part of the input and cannot be changed. The M2LRacS problem is known to be NP-hard in the no-fixed-layer setting (Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2011) 2-Layer Right Angle Crossing Drawings. Proc. IWOCA 2011, Lecturer Notes in Computer Science 7056, pp. 156-169; Di Giacomo, E., Didimo, W., Eades, P. and Liotta, G. (2014) 2-layer right angle crossing drawings. Algorithmica, 68, 954-997), but no algorithm has been proposed so far to solve it. We prove that the M2LRacS problem remains NP-hard even in the 1-?xed-layer setting, and provide different heuristics to solve it in the two settings; one of these heuristics is a 3-approximation algorithm for the no-fixed-layer setting. Also, we present the results of an experimental study that compares our heuristics and shows the effectiveness of the 3-approximation algorithm in practice. [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]

A 2-layer drawing represents a bipartite graph where each vertex is a point on one of two parallel lines, no two vertices on the same line are adjacent, and the edges are straight-line segments. In this paper we study 2-layer drawings where any two crossing edges meet at right angle. We characterize the graphs that admit this type of drawing, provide linear-time testing and embedding algorithms, and present a polynomial-time crossing minimization technique. Also, for a given graph G and a constant k, we prove that it is $\mathcal{NP}$-complete to decide whether G contains a subgraph of at least k edges having a 2-layer drawing with right angle crossings. [ABSTRACT FROM AUTHOR]

Abstract: Let S be a set of N grid points in the plane, no two of which lie on the same horizontal or vertical line, and let G be a graph with n vertices ( ). An orthogeodesic point-set embedding of G on S is a drawing of G such that each vertex is drawn as a point of S and each edge is a chain of horizontal and vertical segments with bends on grid points whose length is equal to the Manhattan distance of its end vertices. We study the following problem. Given a family of trees, what is the minimum value such that every n-vertex tree in admits an orthogeodesic point-set embedding on every set of grid points of size such that no two points lie on the same horizontal or vertical line? We provide polynomial upper bounds on for both planar and non-planar orthogeodesic point-set embeddings as well as for the case when edges are required to be L-shaped. [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: We introduce and study -EMST drawings, i.e., planar straight-line drawings of trees such that, for any fixed , the distance between any two vertices is at least the length of the longest edge in the path connecting them. -EMST drawings are good approximations of Euclidean minimum spanning trees. While it is known that only trees with bounded degree have a Euclidean minimum spanning tree realization, we show that every tree T has a -EMST drawing for any given . We also present drawing algorithms that compute -EMST drawings of trees with bounded degree in polynomial area. As a byproduct of one of our techniques, we improve the best known area upper bound for Euclidean minimum spanning tree realizations of complete binary trees. [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]

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k). [ABSTRACT FROM AUTHOR]

Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts. We exploit the interplay between these notions to describe colored sets of points and to design polynomial-time algorithms to embed k-colored planar graphs on these sets such that the resulting drawings have $\mathcal{O}(k)$ bends per edge. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position. [Copyright &y& Elsevier]

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: This paper studies the problem of drawing directed acyclic graphs in three dimensions in the straight-line grid model so that all directed edges are oriented in a common (upward) direction. We show that there exists a family of outerplanar directed acyclic graphs whose volume requirement is super-linear. We also prove that for the case of directed trees a linear-volume upper bound is achievable. [Copyright &y& Elsevier]

Abstract: Let be a planar graph with vertices and with a partition of the vertex set into subsets for some positive integer . Let be a set of distinct points in the plane with a partition into subsets with (). This paper studies the problem of computing a planar polyline drawing of , such that each vertex of is mapped to a distinct point of . Lower and upper bounds on the number of bends per edge are proved for any . In the special case , we improve the upper and lower bounds presented in a paper by Pach and Wenger [J. Pach, R. Wenger, Embedding planar graphs at fixed vertex locations, Graphs and Combinatorics 17 (2001) 717–728]. The upper bound is based on an algorithm for computing a topological book embedding of a planar graph, such that the vertices follow a given left-to-right order and the number of crossings between every edge and the spine is asymptotically optimal, which can be regarded as a result of independent interest. [Copyright &y& Elsevier]

Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle. [ABSTRACT FROM AUTHOR]

Let R and B be two sets of points such that the points of R are colored red and the points of B are colored blue. Let G be a planar graph such that |R| vertices of G are red and |B| vertices of G are blue. A bichromatic point-set embedding of G on R ∪ B is a crossing-free drawing of G such that each blue vertex is mapped to a point of B, each red vertex is mapped to a point of R, and each edge is a polygonal curve. We study the curve complexity of bichromatic point-set embeddings; i.e., the number of bends per edge that are necessary and sufficient to compute such drawings. We show that O(n) bends are sometimes necessary. We also prove that two bends per edge suffice if G is a 2-colored caterpillar and that for properly 2-colored caterpillars, properly 2-colored wreaths, 2-colored paths, and 2-colored cycles the number of bends per edge can be reduced to one, which is worst-case optimal. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates the basic problem of computing crossing-free straight-line 3D grid drawings of graphs such that the overall volume is small. Motivated by their relevance in the literature, we focus on families of graphs having constant queue number and on k-trees. We present algorithms that compute drawings of these families of graphs on 3D grids consisting of a constant number of parallel lines and such that the overall volume is linear. Lower bounds on the number of such grid lines are also provided. Our results extend and improve similar ones already described in the literature. [Copyright &y& Elsevier]

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ∈C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented. [Copyright &y& Elsevier]

The planar slope numberpsn(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{psn}\,}}(G)$$\end{document} of a planar graph G is the minimum number of edge slopes in a planar straight-line drawing of G. It is known that psn(G)∈O(cΔ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{psn}\,}}(G) \in O(c^{\Delta })$$\end{document} for every planar graph G of maximum degree Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document}. This upper bound has been improved to O(Δ5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\Delta ^5)$$\end{document} if G has treewidth three, and to O(Δ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\Delta )$$\end{document} if G has treewidth two. In this paper we prove psn(G)≤max{4,Δ}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\,\textrm{psn}\,}}(G) \le \max \{4,\Delta \}$$\end{document} when G is a Halin graph, and thus has treewidth three. Furthermore, we present the first polynomial upper bound on the planar slope number for a family of graphs having treewidth four. Namely we show that O(Δ2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\Delta ^2)$$\end{document} slopes suffice for nested pseudotrees. [ABSTRACT FROM AUTHOR]

It is proved that every series-parallel digraph whose maximum vertex degree is Δ admits an upward planar drawing with at most one bend per edge such that each edge segment has one of Δ distinct slopes. The construction is worst-case optimal in terms of the number of slopes, and it gives rise to drawings with optimal angular resolution π Δ. A variant of the drawing algorithm is used to show that (non-directed) reduced series-parallel graphs and flat series-parallel graphs have a (non-upward) 1-bend planar drawing with ⌈ Δ 2 ⌉ distinct slopes if biconnected, and with ⌈ Δ 2 ⌉ + 1 distinct slopes if connected. [ABSTRACT FROM AUTHOR]

In a visibility representation of a graph G , the vertices are represented by non-overlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G , the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given. [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]

Content-based travel recommender systems suggest touristic attractions based on a best match between users’ preferences and a given set of points of interests, called POIs for short. When designing such systems, a critical aspect is to equip them with a rich enough knowledge base that, for each POI, indicates how much the POI is relevant for a set of possible topics of interests, also called TOIs for short. This paper focuses on the problem of designing the Content Analyzer of a content-based travel recommender system. The Content Analyzer is a module that receives as input a set of POIs and a set of TOIs and it computes the relevance of each POI with respect to each TOI. The proposed approach is unsupervised, fully automatic, and it relies on publicly available sources of information. We describe an implementation of the technique in a system called Cicero and present an experimental evaluation of its effectiveness against a ground truth generated by experts. [ABSTRACT FROM AUTHOR]