A storyline visualization is a layout that represents the temporal dynamics of social interactions along time by the convergence of chronological lines. Among the criteria oriented at improving aesthetics and legibility of a representation of this type, a small number of line crossings is the hardest to achieve. We model the crossing minimization in the storyline visualization problem as a multi-layer crossing minimization problem with tree constraints. Our algorithm can compute a layout with the minimum number of crossings of the chronological lines. Computational results demonstrate that it can solve instances with more than 100 interactions and with more than 100 chronological lines to optimality.

A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of G is the minimum number of queues required by any queue layout of G.In this paper, we continue the study of the queue number of planar 3trees. As opposed to general planar graphs, whose queue number is not known to be bounded by a constant, the queue number of planar 3-trees has been shown to be at most seven. In this work, we improve the upper bound to five. We also show that there exist planar 3-trees, whose queue number is at least four; this is the first example of a planar graph with queue number greater than three.

Back in the eighties, Heath [6] showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem.Lemma 2. Given a 4-planar triconnected graph G and a separating trianglepairwise distinct or all represent the same vertex. Proof. In the other case, where w.l.o.g. A in = B in = v and Γ in = v, there exists a separation pair (v, Γ) contradicting the triconnectivity of G. A symmetric argument applies to A out , B out , Γ out . Lemma 3. In a 4-planar triconnected graph, every pair of distinct separating triangles T and T is vertex disjoint, i.e. V (T ) ∩ V (T ) = ∅.Proof. Assume to the contrary that T and T share an edge or a vertex. In the first case, let w.l.o.g. e = (u, v) be the common edge. The degree of both u and v is at least five, since three edges are required for T , T and two additional edges to connect G in (T ) and G in (T ) to T and T , respectively. In the second case, let v denote the common vertex. Since v is part of two edge disjoint cycles and connected to G in (T ) and G in (T ), it follows that deg(v) ≥ 6.Consider now a 4-planar triconnected graph with a single separating triangle T . Similar to Chen [2], the idea is to compute two cycles H in (T ) and H out (T ) for G in (T ) and G out (T ) and link them via the separating triangle together. The crucial observation is that if two cycles intersect as illustrated in Fig. 2, i.e., they contain two edges of the triangle but have only one of them in common, then we can always merge them into one cycle. Lemma 4. Let G be a triconnected 4-planar graph, T a separating triangle, and H in (T ) and H out (T ) two subhamiltonian cycles for G in (T ) and G out (T ), resp. If E(H in (T )) ∩ E(T ) = {e in , e} and E(H out (T )) ∩ E(T ) = {e out , e} where {e, e in , e out } are the edges of T , then G is subhamiltonian. Proof. Let w.l.o.g. e = (A, B), e in = (B, Γ) and e out = (A, Γ) as illustrated in Fig. 2. The result of removing the edges of T from both cycles are two paths P out = B Γ and P in = Γ A. Joining them at Γ and inserting e yields a subhamiltonian cycle.

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In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal (45• ) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A k-planar graph is a planar graph in which each vertex has degree less or equal to k. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size O(n 2 ) × O(n). For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge.

Canonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While st-orderings do extend to directed graphs, especially planar st-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic st-orderings to directed graphs. We fully characterize planar st-graphs that admit such an ordering and provide a lineartime algorithm for recognition and ordering. If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on n vertices by using at most one bend per edge, at most n − 3 bends in total and within quadratic area.

Abstract. The visualization of clustered graphs is an essential tool for the analysis of networks, in particular, social networks, in which clustering techniques like community detection can reveal various structural properties. In this paper, we show how clustered graphs can be drawn as topographic maps, a type of map easily understandable by users not familiar with information visualization. Elevation levels of connected entities correspond to the nested structure of the cluster hierarchy. We present methods for initial node placement and describe a tree mapping based algorithm that produces an area efficient layout. Given this layout, a triangular irregular mesh is generated that is used to extract the elevation data for rendering the map. In addition, the mesh enables the routing of edges based on the topographic features of the map.

In the classical Steiner tree problem, given an undirected, connected graph G = (V , E) with non-negative edge costs and a set of terminals T ⊆ V , the objective is to find a minimum-cost tree E ′ ⊆ E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of ρ = ln(4) + ε < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T ℓ ⊂ · · · ⊂ T 1 ⊆ V , compute nested trees E ℓ ⊆ · · · ⊆ E 1 ⊆ E that span the corresponding terminal sets with minimum total cost.The MLST problem and variants thereof have been studied under various names including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two simple O(ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present various integer linear programming (ILP) formulations for the MLST problem, and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ = 100 levels, which is sufficient for most applications such as network visualization or designing multi-level infrastructure.Let G = (V , E) be an undirected, connected graph with positive edge costs c : E → R + , and let T ⊆ V be a set of vertices called terminals. A Steiner tree is a tree in G that spans T . The network (graph) Steiner tree problem (ST) is to find a minimum-cost Steiner tree E ′ ⊆ E, where the cost of E ′ is c(E ′ ) = e ∈E ′ c(e). ST is one of Karp's initial NP-hard problems [12]; see also a survey [21], an online compendium [11], and a textbook [18].Due to its practical importance in many domains, there is a long history of exact and approximation algorithms for the problem. The classical 2-approximation algorithm for ST [10] uses the metric closure of G, i.e., the complete edge-weighted graph G * with vertex set T in which, for every edge uv, the cost of uv equals the length of a shortest u-v path in G. A minimum spanning tree of G * corresponds to a 2-approximate Steiner tree in G.Currently, the last in a long list of improvements is the LP-based approximation algorithm of Byrka et al. [5], which has a ratio of ln(4) + ε < 1.39. Their algorithm uses a new iterative randomized rounding technique. Note that ST is APX-hard [4]; more concretely, it is NP-hard to approximate the problem within a factor of 96/95 [7]. This is in contrast to the geometric variant of the problem, where terminals correspond to points in the Euclidean or rectilinear plane. Both variants admit polynom...