computer science publication server: No conditions. Results ordered -Date Deposited. 2021-04-22T23:30:09Z EPrints http://e-archive.informatik.uni-koeln.de/images/sitelogo.png http://e-archive.informatik.uni-koeln.de/ 2018-03-12T14:18:42Z 2018-03-12T15:06:53Z http://e-archive.informatik.uni-koeln.de/id/eprint/922 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/922 2018-03-12T14:18:42Z Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings Subject of this work are two problems related to ordering the vertices of planar graphs. The first one is concerned with the properties of vertex-orderings that serve as a basis for incremental drawing algorithms. Such a drawing algorithm usually extends a drawing by adding the vertices step-by-step as provided by the ordering. In the field of graph drawing several orderings are in use for this purpose. Some of them, however, lack certain properties that are desirable or required for classic incremental drawing methods. We narrow down these properties, and introduce the bitonic st-ordering, an ordering which combines the features only available when using canonical orderings with the flexibility of st-orderings. The additional property of being bitonic enables an st-ordering to be used in algorithms that usually require a canonical ordering. With this in mind, we describe a linear-time algorithm that computes such an ordering for every biconnected planar graph. Unlike canonical orderings, st-orderings extend to directed graphs, in particular planar st-graphs. Being able to compute bitonic st-orderings for planar st-graphs is of particular interest for upward planar drawing algorithms, since traditional incremental algorithms for undirected planar graphs might be adapted to directed graphs. Based on this observation, we give a full characterization of the class of planar st-graphs that admit such an ordering. This includes a linear-time algorithm for recognition and ordering. Furthermore, we show that by splitting specific edges of an instance that is not part of this class, one is able to transform it into one for which then such an ordering exists. To do so, we describe a linear-time algorithm for finding the smallest set of edges to split. We show that for a planar st-graph G=(V,E), |V|−3 edge splits are sufficient and every edge is split at most once. This immediately translates to the number of bends required for upward planar poly-line drawings. More specifically, we show that every planar st-graph admits an upward planar poly-line drawing in quadratic area with at most |V|−3 bends in total and at most one bend per edge. Moreover, the drawing can be obtained in linear time. The second part is concerned with embedding planar graphs with maximum degree three and four into books. Besides providing a simplified incremental linear-time algorithm for embedding triconnected 3-planar graphs into a book of two pages, we describe a linear-time algorithm to compute a subhamiltonian cycle in a triconnected 4-planar graph. Martin Gronemann Gronemann, Martin 2010-03-15T00:00:00Z 2012-01-09T15:38:58Z http://e-archive.informatik.uni-koeln.de/id/eprint/530 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/530 2010-03-15T00:00:00Z Semi-preemptive routing on a linear and circular track The problem of routing a robot (or vehicle) between n stations in the plane in order to transport objects is well studied, even if the stations are specially arranged, e.g. on a linear track or circle. The robot may use either all or none of the stations for reloading. We will generalize these concepts of preemptiveness/nonpreemptiveness and emancipate the robot by letting it choose k le n reload-stations. We will show that the problem on the linear and circular track remains polynomial solvable. Dirk Räbiger Räbiger, Dirk Rainer Schrader Schrader, Rainer 2009-04-19T00:00:00Z 2012-01-09T16:21:32Z http://e-archive.informatik.uni-koeln.de/id/eprint/511 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/511 2009-04-19T00:00:00Z Semi-Preemptive Routing on Trees We study a variant of the pickup-and-delivery problem (PDP) in which the objects that have to be transported can be reloaded at most d times, for a given integer d. This problem is known to be polynomially solvable on paths or cycles and NP-complete on trees. We present a (4/3+epsilon)-approximation algorithm if the underlying graph is a tree. By using a result of Charikar et al. (1998), this can be extended to a O(log n log log n)-approximation for general graphs. Sven Krumke Krumke, Sven Dirk Räbiger Räbiger, Dirk Rainer Schrader Schrader, Rainer 2007-01-03T00:00:00Z 2012-01-12T10:58:54Z http://e-archive.informatik.uni-koeln.de/id/eprint/475 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/475 2007-01-03T00:00:00Z Semi-preemptive routing on a linear and circular track Please refer to the more recent version of this paper, zaik2007-530. Dirk Räbiger Räbiger, Dirk 2005-08-11T00:00:00Z 2011-12-19T09:44:52Z http://e-archive.informatik.uni-koeln.de/id/eprint/487 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/487 2005-08-11T00:00:00Z Semi-Präemptives Transportieren We consider the problem of routing a robot (or vehicle) between n stations in order to transport m objects from their source node to their destination node such that the total travel distance is minimized. This problem is known as Pickup and Delivery problem and well studied in literature, even if the stations are specially arranged, e.g. on a linear track or circle. The robot may use either all or none of the stations for reloading. These cases are said to be preemptive resp. non--preemptive. We will generalize these concepts by restricting the nodes that can be used for reloading. According to the previous versions we will call this variation semi--preemptive. We will distinguish between an exogenous and an endogenous case. Problems of the first type will allow the robot to use only a subset of stations that can be used as reload nodes. The latter case consists of a number k and it will be part of the problem to decide at which stations the robot reloads, but no more than k times. We will show that both the exogenous and the endogenous problem are efficiently solvable by dynamic programming on paths and circles. Both variants are NP --complete on trees. For the exogenous case we will give an approximation algorithm with ratio 4/3 . The endogenous case can be solved with ratio (4/3+c) , for arbitrary c>0 . In conclusion we use a known result in order to approximately solve the exogenous resp. endogenous routing problem on graphs weighted by a metric. We will study a graph theoretical problem which arises from the mentioned transportation problem. Given a directed graph G=(V,E^rdotcup E^b) with the arc set partitioned into red and blue subsets and a cost function on the arc set, find a minimal cost arborescence spanning G and using at most d blue arcs. Within this work a fully polynomial time approximation scheme (FPTAS) will be presented for this problem. The FPTAS will be used to approximately solve the endogenous transportation problem on a tree. Dirk Räbiger Räbiger, Dirk 2001-04-02T00:00:00Z 2011-10-24T09:04:24Z http://e-archive.informatik.uni-koeln.de/id/eprint/43 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/43 2001-04-02T00:00:00Z On a generalization of a theorem of Nash-Williams Bert Faßbender Faßbender, Bert 2001-04-02T00:00:00Z 2012-01-19T11:30:49Z http://e-archive.informatik.uni-koeln.de/id/eprint/98 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/98 2001-04-02T00:00:00Z On the Complexity of the Disjoint Path Problem We consider the disjoint paths problem. Given a graph G and a subset S of the edge-set of G the problem is to decide whether there exists a family F of disjoint circuits in G each containing exactly one edge of S such that every edge in S belongs to a circuit in C. By a well-known theorem of P. Seymour [On odd cuts and plane multicommodity flows, Proc. London Math. Soc.(3) 42, 178-192 (1981)] the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphs G. We show that (assuming P e NP) one can drop neither planarity nor the Eulerian condition on G without losing polynomial time solvability. We prove the NP-completeness of the planar edge-disjoint paths problem by showing the NP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assuming P e NP) a conjecture of A. Schrijver [Homotopic Routing Methods, in: Paths, Flows and VLSI Layout, Algorithms Comb. 9, 329-371 (1990)] concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjugate of A. Frank mentioned in A. Seboe [Dual Integrality and Multicommodity Flows. Combinatorics, Colloquia Mathematica Societatis Janos Bolyai, 52, 453-469 (1988)]. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minor-closed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show the NP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs. Matthias Middendorf Middendorf, Matthias Frank Pfeiffer Pfeiffer, Frank 2001-04-02T00:00:00Z 2011-10-21T13:25:49Z http://e-archive.informatik.uni-koeln.de/id/eprint/78 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/78 2001-04-02T00:00:00Z A Sufficient Condition on Degree Sums of Independent Triples for Hamiltonian Cycles in 1-Tough Graphs Bert Faßbender Faßbender, Bert 2001-04-02T00:00:00Z 2011-10-24T14:10:48Z http://e-archive.informatik.uni-koeln.de/id/eprint/46 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/46 2001-04-02T00:00:00Z Kriterien vom Ore-Typ für längste Kreise in 2-zusammenhängenden Graphen Let G be a simple 2-connected graph of order u such that the degree-sum of any two nonadjacent vertices is at least {2 over 3} u, and let C be a non-dominating longest cycle in G. We show that G-C is a complete graph and V(G) contains a nonempty proper subset S such that G-S has exactly |S|+1 components. From this result a sufficient condition for the existence of Hamilton cycles in 1-tough graphs can be obtained. Bert Faßbender Faßbender, Bert 2001-04-02T00:00:00Z 2011-12-19T09:46:20Z http://e-archive.informatik.uni-koeln.de/id/eprint/281 This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/281 2001-04-02T00:00:00Z Large Circuits in Binary Matroids of Large Cogirth: I Let F7 denote the Fano matroid and e be a fixed element of F7. Let P(F7,e) be the family of matroids obtained by taking the parallel connection of one or more copies of F7 about e. Let M be a simple binary matroid such that every cocircuit of M has size at least d >= 3. We show that if M does not have an F7-minor, M is not F * 7 and d >= (r(M)+1)/2 then M has a circuit of size r(M)+1. We also show that if M is connected, e in E(M), M does not have both an F7-minor and an F * 7-minor, and M is not in P(F7,e), then M has a circuit containing e and of size at least d+1. Winfried Hochstättler Hochstättler, Winfried Bill Jackson Jackson, Bill