Metro maps are schematic diagrams of public transport networks that serve as visual aids for route planning and navigation tasks. It is a challenging problem in network visualization to automatically draw appealing metro maps. There are two aspects to this problem that depend on each other: the layout problem of finding station and link coordinates and the labeling problem of placing non-overlapping station labels. par In this paper we present a new integral approach that solves the combined layout and labeling problem (each of which, independently, is known to be NP-hard) using mixed-integer programming (MIP). We identify seven design rules used in most real-world metro maps. We split these rules into hard and soft constraints and translate them into a MIP model. Our MIP formulation finds a metro map that satisfies all hard constraints (if such a drawing exists) and minimizes a weighted sum of costs that correspond to the soft constraints. We have implemented the MIP model and present a case study and the results of an expert assessment to evaluate the performance of our approach in comparison to both manually designed official maps and results of previous layout methods.

For $t>0$ and $gge 0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for every constant $t>0$, there is a constant $gge 0$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling.

In this paper we present algorithms for computing large matchings in 3-regular graphs, graphs with maximum degree 3, and 3-connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the best-known algorithm for computing maximum matchings in general graphs, which runs in $O(nm)$ time, where $n$ denotes the number of vertices and $m$ the number of edges of the given graph. For the classes of 3-regular graphs and graphs with maximum degree 3, the bounds we achieve are known to be best possible. par We also investigate graphs with block trees of bounded degree, where the $d$-block tree is the adjacency graph of the $d$-connected components of the given graph. In 3-regular graphs and 3-connected planar graphs with bounded-degree 2- and 4-block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.

A straight-line drawing $ of a planar graph $G$ need not be plane, but can be made so by untangling it, that is, by moving some of the vertices of $G$. Let shift$(G,$ denote the minimum number of vertices that need to be moved to untangle $. We show that shift$(G,$ is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem. par Further we define fix$(G,=n-shift(G,$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $. We give an algorithm that fixes at least $((log n)-1)/log log n$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $n/2$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $G$ of $G$ with fix$(G,G) le n-2+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $H$ of $H$ with fix$(H,H) le 2 n-1+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

We study the problem of morphing between two polylines that represent linear geographical features like roads or rivers generalized at two different scales. This problem occurs frequently during continuous zooming in interactive maps. Situations in which generalization operators like typification and simplification replace, for example, a series of consecutive bends by fewer bends are not always handled well by traditional morphing algorithms. We attempt to cope with such cases by modeling the problem as an optimal correspondence problem between characteristic parts of each polyline. A dynamic programming algorithm is presented that solves the matching problem in $O(nm)$ time, where $n$ and $m$ are the respective numbers of characteristic parts of the two polylines. In a case study we demonstrate that the algorithm yields good results when being applied to data from mountain roads, a river and a region boundary at various scales.

This paper discusses algorithms for labeling sets of points in the plane, where labels are not restricted to some finite number of positions. We show that continuously sliding labels allow more points to be labeled both in theory and in practice. We define six different models of labeling. We compare models by analyzing how many more points can receive labels under one model than another. We show that maximizing the number of labeled points is NP-hard in the most general of the new models. Nevertheless, we give a polynomial-time approximation scheme and a simple and efficient factor-1/2 approximation algorithm for each of the new models. par Finally, we give experimental results based on the factor-1/2 approximation algorithm to compare the models in practice. We also compare this algorithm experimentally to other algorithms suggested in the literature.