Papers by Michael Kaufmann
Algorithmica, 2016
In a book embedding, the vertices of a graph are placed on the "spine" of a book and the edges ar... more In a book embedding, the vertices of a graph are placed on the "spine" of a book and the edges are assigned to "pages", so that edges on the same page do not cross. In this paper, we prove that every 1-planar graph (that is, a graph that can be drawn on the plane such that no edge is crossed more than once) admits an embedding in a book with constant number of pages. To the best of our knowledge, the best non-trivial previous upper-bound is O(√ n), where n is the number of vertices of the graph.
Lecture Notes in Computer Science, 1997
Graph drawing algorithms usually attempt to display the characteristic properties of the input gr... more Graph drawing algorithms usually attempt to display the characteristic properties of the input graphs. In this paper we consider the class of planar bipartite graphs and try to achieve planar drawings such that the bipartiteness property is cleary shown. To this aim, we develop several models, give e cient algorithms to nd a corresponding drawing if possible or prove the hardness of the problem.
Lecture Notes in Computer Science, 1995
... The Euler-tour technique requires the application of several subroutines, one of which islist... more ... The Euler-tour technique requires the application of several subroutines, one of which islist ranking. Contents. ... For PRAM algorithms, it is commonly assumed that the graph is given either in the form of an adjacency list or as an adjacency matrix. ...
Lecture Notes in Computer Science, 1994
The rectilinear Steiner tree problem requires to find a shortest tree connecting a given set of t... more The rectilinear Steiner tree problem requires to find a shortest tree connecting a given set of terminal points in the plane with rectilinear distance. We show that the performance ratios of Zelikovsky's [17] heuristic is between 1.3 and 1.3125 (before it was only bounded from ...
Lecture Notes in Computer Science, 2013
In this paper we study threshold coloring of graphs, where the vertex colors represented by integ... more In this paper we study threshold coloring of graphs, where the vertex colors represented by integers are used to describe any spanning subgraph of the given graph as follows. Pairs of vertices with near colors imply the edge between them is present and pairs of vertices with far colors imply the edge is absent. Not all planar graphs are threshold-colorable, but several subclasses, such as trees, some planar grids, and planar graphs without short cycles can always be threshold-colored. Using these results we obtain unit-cube contact representation of several subclasses of planar graphs. Variants of the threshold coloring problem are related to well-known graph coloring and other graph-theoretic problems. Using these relations we show the NP-completeness for two of these variants, and describe a polynomial-time algorithm for another.
Lecture Notes in Computer Science, 2015
Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring p... more Given an n-vertex graph G and two positive integers d, k ∈ N, the (d, kn)-differential coloring problem asks for a coloring of the vertices of G (if one exists) with distinct numbers from 1 to kn (treated as colors), such that the minimum difference between the two colors of any adjacent vertices is at least d. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NPcomplete, our main contribution is a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. For practical reasons, we also consider color ranges larger than n, i.e., k > 1. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the (2n/3 , 2n)-differential coloring problem, even in the case where the input graph is planar.
Lecture Notes in Computer Science, 2013
In this paper we study many-to-one boundary labeling with backbone leaders. In this model, a hori... more In this paper we study many-to-one boundary labeling with backbone leaders. In this model, a horizontal backbone reaches out of each label into the feature-enclosing rectangle. Feature points associated with this label are linked via vertical line segments to the backbone. We present algorithms for label number and leader-length minimization. If crossings are allowed, we aim to minimize their number. This can be achieved efficiently in the case of fixed label order. We show that the corresponding problem in the case of flexible label order is NP-hard.
Journal of Discrete Algorithms, 2014
ABSTRACT We study the maximum differential coloring problem, where the vertices of an n-vertex gr... more ABSTRACT We study the maximum differential coloring problem, where the vertices of an n-vertex graph must be labeled with distinct numbers ranging from 1 to n, so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is NP-hardfor general graphs [16], we consider planar graphs and subclasses thereof. We prove that the maximum differential coloring problem remains NP-hard, even for planar graphs. We also present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Miller-Pritikin labeling scheme [19] for forests is optimal for regular caterpillars and for spider graphs.
The Computer Journal, 2009
Boundary labeling is a relatively new labeling method. It can be useful in automating the product... more Boundary labeling is a relatively new labeling method. It can be useful in automating the production of technical drawings and medical maps, where it is common to explain certain parts of the drawing with text labels, arranged on its boundary so that other parts of the drawing are not obscured. In boundary labeling, we are given a rectangle R which encloses a set of n sites. Each site si is associated with an axis-parallel rectangular label li. The labels must be placed in distinct positions on the boundary of R and to be connected to their corresponding sites with polygonal lines, called leaders, so that the labels are pairwise disjoint and the leaders do not intersect each other. In this paper, we study a version of the boundary labeling problem where the sites can "float" within a polygonal region. We present a polynomial time algorithm that produces a labeling of minimum total leader length for labels of uniform size placed in fixed positions on the boundary of R.
Lecture Notes in Computer Science, 2011
We study the problem of characterizing the directed graphs with an upward straight-line embedding... more We study the problem of characterizing the directed graphs with an upward straight-line embedding into every point set in general or in convex position. We solve two questions posed by Binucci et al. [Computational Geometry: Theory and Applications, 2010]. Namely, we prove that the classes of directed graphs with an upward straight-line embedding into every point set in convex position and with an upward straight-line embedding into every point set in general position do not coincide, and we prove that every directed caterpillar admits an upward straight-line embedding into every point set in convex position. Further, we provide new partial positive results on the problem of constructing upward straight-line embeddings of directed paths into point sets in general position.
Lecture Notes in Computer Science, 2014
Graph and cartographic visualization have the common objective to provide intuitive understanding... more Graph and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible on a map with rectangular regions. We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.
We study representations of graphs by contacts of circular arcs, CCA-representations for short, w... more We study representations of graphs by contacts of circular arcs, CCA-representations for short, where the vertices are interior-disjoint circular arcs in the plane and each edge is realized by an endpoint of one arc touching the interior of another. A graph is (2, k)-sparse if every s-vertex subgraph has at most 2s − k edges, and (2, k)-tight if in addition it has exactly 2n − k edges, where n is the number of vertices. Every graph with a CCA-representation is planar and (2, 0)-sparse, and it follows from known results on contacts of line segments that for k ≥ 3 every (2, k)-sparse graph has a CCA-representation. Hence the question of CCA-representability is open for (2, k)-sparse graphs with 0 ≤ k ≤ 2. We partially answer this question by computing CCA-representations for several subclasses of planar (2, 0)-sparse graphs. In particular, we show that every plane (2, 2)-sparse graph has a CCA-representation, and that any plane (2, 1)-tight graph or (2, 0)-tight graph dual to a (2, 3)-tight graph or (2, 4)-tight graph has a CCA-representation. Next, we study CCA-representations in which each arc has an empty convex hull. We characterize the plane graphs that have such a representation, based on the existence of a special orientation of the graph edges. Using this characterization, we show that every plane graph of maximum degree 4 has such a representation, but that finding such a representation for a plane (2, 0)-tight graph with maximum degree 5 is an NP-complete problem. Finally, we describe a simple algorithm for representing plane (2, 0)-sparse graphs with wedges, where each vertex is represented with a sequence of two circular arcs (straight-line segments).
Lecture Notes in Computer Science, 2011
We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a... more We study the problem of Upward Point-Set Embeddability, that is the problem of deciding whether a given upward planar digraph D has an upward planar embedding into a point set S. We show that any switch tree admits an upward planar straight-line embedding into any convex point set. For the class of k-switch trees, that is a generalization of switch trees (according to this definition a switch tree is a 1-switch tree), we show that not every k-switch tree admits an upward planar straightline embedding into any convex point set, for any k ≥ 2. Finally we show that the problem of Upward Point-Set Embeddability is NP-complete.
Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06, 2006
Max-tolerance graphs can be regarded as generalized interval graphs, where two intervals I i and ... more Max-tolerance graphs can be regarded as generalized interval graphs, where two intervals I i and I j only induce an edge in the corresponding graph iff they overlap for an amount of at least max{t i , t j } where t i is an individual tolerance parameter associated to each interval I i. A new geometric characterization of max-tolerance graphs as intersection graphs of isosceles right triangles, shortly called semi-squares, leverages the solution of various graphtheoretic problems in connection with max-tolerance graphs. First, we solve the maximal and maximum cliques problem. It arises naturally in DNA sequence analysis where the maximal cliques might be interpreted as functional domains carrying biologically meaningful information. We prove an upper bound of O(n 3) for the number of maximal cliques in max-tolerance graphs and give an efficient O(n 3) algorithm for their computation. In the same vein, the semi-square representation yields a simple proof for the fact that this bound is asymptotically tight, i.e., a class of max-tolerance graphs is presented where the instances have Ω(n 3) maximal cliques. Additionally, we answer an open question posed in [8] by showing that max-tolerance graphs do not contain complements of cycles C n for n > 9. By exploiting the new representation more deeply, we can go even further and prove that the recognition problem for max-tolerance graphs is NP-hard.
Journal of Graph Algorithms and Applications, 2014
A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges a... more A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph G. In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether it admits an orientation resulting in a directed plane graph G such that G admits a planar drawing in which each edge is represented by a curve monotonically increasing in the y-direction according to its orientation. Our contribution is threefold. First, we show that the upward planarity testing problem is solvable in cubic time for mixed outerplane graphs. Second, we show that the problem of testing the upward planarity of mixed plane graphs reduces in quadratic time to the problem of testing the upward planarity of mixed plane triangulations. Third, we exhibit linear-time testing algorithms for two classes of mixed plane triangulations, namely mixed plane 3-trees and mixed plane triangulations in which the undirected edges induce a forest.
Journal of Graph Algorithms and Applications, 2012
Two graphs G1 = (V, E1) and G2 = (V, E2) admit a geometric simultaneous embedding if there exist ... more Two graphs G1 = (V, E1) and G2 = (V, E2) admit a geometric simultaneous embedding if there exist a set of points P and a bijection M : V → P that induce planar straight-line embeddings both for G1 and for G2. The most prominent problem in this area is the question of whether a tree and a path can always be simultaneously embedded. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. Finally, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of height 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution,
Journal of Graph Algorithms and Applications, 2009
A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective... more A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ : V1 → V2 is a pair of planar straightline drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2. In this paper we examine several constrained versions of the geometric simultaneous embedding problem as well as a more relaxed version in which instead of exactly simultaneous we look for near-simultaneous embeddings. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, we show that if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the nearsimultaneous embedding problem, in which vertices are not forced to be placed exactly in the same, but just in "near" points in different drawings.
Journal of Graph Algorithms and Applications, 2005
Sugiyama's algorithmic framework for layered graph drawing is commonly used in practical software... more Sugiyama's algorithmic framework for layered graph drawing is commonly used in practical software. The extensive use of dummy vertices to break long edges between non-adjacent layers often leads to unsatisfactorial performance. The worst-case running-time of Sugiyama's approach is O(|V ||E| log |E|) requiring O(|V ||E|) memory, which makes it unusable for the visualization of large graphs. By a conceptually simple new technique we are able to keep the number of dummy vertices and edges linear in the size of the graph and hence reduce the worst-case time complexity of Sugiyama's approach by an order of magnitude to O((|V | + |E|) log |E|) requiring O(|V | + |E|) space.
Journal of Graph Algorithms and Applications, 2002
The existing literature gives efficient algorithms for mapping trees or less restrictively outerp... more The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.
Journal of Graph Algorithms and Applications, 2013
In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs t... more In this paper, we study the geometric RAC simultaneous drawing problem: Given two planar graphs that share a common vertex set but have disjoint edge sets, a geometric RAC simultaneous drawing is a straight-line drawing in which (i) each graph is drawn planar, (ii) there are no edge overlaps, and, (iii) crossings between edges of the two graphs occur at right-angles. We first prove that two planar graphs admitting a geometric simultaneous drawing may not admit a geometric RAC simultaneous drawing. We further show that a cycle and a matching always admit a geometric RAC simultaneous drawing, which can be constructed in linear time. We also study a closely related problem according to which we are given a planar embedded graph G and the main goal is to determine a geometric drawing of G and its dual G * (without the face-vertex corresponding to the external face) such that: (i) G and G * are drawn planar, (ii) each vertex of the dual is drawn inside its corresponding face of G and, (iii) the primal-dual edge crossings form right-angles. We prove that it is always possible to construct such a drawing if the input graph is an outerplanar embedded graph.
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Papers by Michael Kaufmann