computer science publication server: No conditions. Results ordered -Date Deposited. 2021-04-22T23:55:20ZEPrintshttp://e-archive.informatik.uni-koeln.de/images/sitelogo.pnghttp://e-archive.informatik.uni-koeln.de/2012-12-04T13:37:55Z2013-07-08T09:55:53Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/687This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/6872012-12-04T13:37:55ZOn the separability of graphsRecently, Cicalese and Milanič introduced a graph-theoretic concept called separability. A graph is said to be k-separable if any two non-adjacent vertices can be separated by the removal of at most k vertices. The separability of a graph G is the least k for which G is k-separable. In this paper, we investigate this concept under the following three aspects. First, we characterize the graphs for which in any non-complete connected induced subgraph the connectivity equals the separability, so-called separability-perfect graphs. We list the minimal forbidden induced
subgraphs of this condition and derive a complete description of the
separability-perfect graphs.We then turn our attention to graphs for which the separability is given locally by the maximum intersection of the neighborhoods of any two non-adjacent vertices. We prove that all (house,hole)-free graphs fulfill this property – a class properly including the chordal graphs and the distance-hereditary graphs. We conclude that the separability can be computed in O(m∆) time for such graphs.In the last part we introduce the concept of edge-separability, in analogy to edge-connectivity, and prove that the class of k-edge-separable graphs is closed under topological minors for any k. We explicitly give the forbidden topological minors of the k-edge-separable graphs for each 0 ≤ k ≤ 3.
Oliver SchaudtSchaudt, OliverRainer SchraderSchrader, RainerVera WeilWeil, Vera2011-12-05T12:55:29Z2014-08-12T14:14:50Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/633This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/6332011-12-05T12:55:29ZOn separation pairs and split components of biconnected graphsThe decomposition of a biconnected graph G into its triconnected components is fundamental in graph theory and has a wide range of applications. Based on a palm tree of G, the algorithm by Hopcroft and Tarjan is able to compute them in linear time if some corrections are applied.
Today, the algorithm is still considered very hard to understand and proofs of its correctness are technical and challenging.
The article at hand provides a more comprehensive description of the algorithm, making it easier to understand and implement. Its correctness is validated by explicitly mapping the algorithmic detection criteria to
the graph-theoretic characterization of type-1 and type-2 separation pairs.
Further, it reveals further errors and inaccuracies in the common definitions. This includes the description and proofs of further properties and relationships of separation pairs. The presented results also answer the question whether and under which preconditions type-1 and type-2 pairs can be computed separately from each other.Sven MallachMallach, Sven2010-08-31T14:50:33Z2014-07-04T09:34:50Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/817This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/8172010-08-31T14:50:33ZOn Partitioning the Edges of Graphs into Connected SubgraphsFor any positive integer s, an s-partition of a graph G = (V, E) is a partition of E into E1 E2 Ek, where Ei = s for 1 i k - 1 and 1 Ek s and each Ei induces a connected subgraph of G. We prove
(i) If G is connected, then there exists a 2-partition, but not necessarily a 3-partition;
(ii) If G is 2-edge connected, then there exists a 3-partition, but not necessarily a 4-partition;
(iii) If G is 3-edge connected, then there exists a 4-partition;
(iv) If G is 4-edge connected, then there exists an s-partition for all s.Michael JüngerJünger, MichaelGerhard ReineltReinelt, GerhardWilliam R. PulleyblankPulleyblank, William R.2007-06-04T00:00:00Z2011-08-11T15:37:59Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/542This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/5422007-06-04T00:00:00ZOn the Weighted Minimal Deletion of Rooted Bipartite MinorsWe investigate the problem of finding a minimal weighted set of edges whose removal results in a graph without minors that are contractible onto a prespecified set of vertices. Such minors are called rooted. The problem of a minimal weighted deletion of all rooted K(i,3)-minors for a fixed integer i is proved to be NP-hard on general graphs. Furthermore, a polynomial time algorithm is developed for the rooted K(1,3)-minor deletion problem on planar graphs while for the rooted K(2,3)-free minor planar graphs a characterization is presented.Elisabeth GassnerGassner, ElisabethMerijam PercanPercan, Merijam2006-02-08T00:00:00Z2012-01-09T16:17:35Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/515This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/5152006-02-08T00:00:00ZOn a relation between the domination number and a strongly connected bidirection of an undirected graphAs a generalization of directed and undirected graphs, Edmonds and Johnson introduced bidirected graphs. A bidirected graph is a graph each arc of which has either two positive end-vertices (tails), two negative end-vertices (heads), or one positive end-vertex (tail) and one negative end-vertex (head). We extend the notion of directed paths, distance, diameter and strong connectivity from directed to bidirected graphs and characterize those undirected graphs that allow a strongly connected bidirection. Considering the problem of finding the minimum diameter of all strongly connected bidirections of a given undirected graph, we generalize a result of Fomin et al. about directed graphs and obtain an upper bound for the minimum diameter which depends on the minimum size of a dominating set and the number of bridges in the undirected graph.Martin LätschLätsch, MartinBritta PeisPeis, Britta2005-12-07T00:00:00Z2012-01-12T11:28:38Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/435This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/4352005-12-07T00:00:00ZSubgraph Induced Connectivity AugmentationGiven a planar graph G=(V,E) and a vertex set Wsubseteq V , the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such that G'=(V,Ecup F) is planar and the subgraph of G' induced by W is connected. The problem arises in automatic graph drawing in the context of c -planarity testing of clustered graphs. We describe a linear time algorithm based on SPQR-trees that tests if a subgraph induced planar connectivity augmentation exists and, if so, constructs an minimum cardinality augmenting edge set.Carsten GutwengerGutwenger, CarstenMichael JüngerJünger, MichaelSebastian LeipertLeipert, SebastianPetra MutzelMutzel, PetraMerijam PercanPercan, MerijamRené WeiskircherWeiskircher, René2003-04-03T00:00:00Z2012-01-12T12:18:06Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/436This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/4362003-04-03T00:00:00ZAdvances in C-Planarity Testing of Clustered Graphs (Extended Abstract)A clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E) . Each vertex c in T corresponds to a subset of the vertices of the graph called ''cluster''. C -planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automatic graph drawing. The complexity status of c -planarity testing is unknown. It has been shown that c -planarity can be tested in linear time for c -connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In this paper, we provide a polynomial time algorithm for c -planarity testing for 'àlmost'' c -connected clustered graphs, i.e., graphs for which all c -vertices corresponding to the non- c -connected clusters lie on the same path in T starting at the root of T , or graphs in which for each non-connected cluster its super-cluster and all its siblings are connected. The algorithm uses ideas of the algorithm for subgraph induced planar connectivity augmentation. We regard it as a first step towards general c -planarity testing.Carsten GutwengerGutwenger, CarstenMichael JüngerJünger, MichaelSebastian LeipertLeipert, SebastianPetra MutzelMutzel, PetraMerijam PercanPercan, MerijamRené WeiskircherWeiskircher, René2002-12-19T00:00:00Z2012-01-12T12:01:09Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/444This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/4442002-12-19T00:00:00ZTriangulating Clustered GraphsA clustered graph C=(G,T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G=(V,E) . Each vertex mu in T corresponds to a subset of the vertices of the graph called ''cluster''. C -planarity is a natural extension of graph planarity for clustered graphs. As we triangulate a planar embedded graph so that G is still planar embedded after triangulation, we consider triangulation of a c -connected clustered graph that preserve the c -planar embedding. In this paper, we provide a linear time algorithm for triangulating c -connected c -planar embedded clustered graphs C=(G,T) so that C is still c -planar embedded after triangulation. We assume that every non-trivial cluster in C has at least two childcluster. This is the first time, this problem was investigated.Michael JüngerJünger, MichaelSebastian LeipertLeipert, SebastianMerijam PercanPercan, Merijam2001-04-02T00:00:00Z2011-10-24T09:04:24Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/43This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/432001-04-02T00:00:00ZOn a generalization of a theorem of Nash-WilliamsBert FaßbenderFaßbender, Bert2001-04-02T00:00:00Z2011-10-24T12:47:56Zhttp://e-archive.informatik.uni-koeln.de/id/eprint/38This item is in the repository with the URL: http://e-archive.informatik.uni-koeln.de/id/eprint/382001-04-02T00:00:00ZOrder-Degree SequencesLet G be a graph with vertices x1,...,xn. The order-degree sequence of G is the maximal n-tuple (a1,...,an)=(a(x1),...,a(xn)) of nonnegative integers such that, for i=1(1)n, vertex xi is joined to distinct vertices y1,...,y{asb i} with a(yj) >= j for j=1(1)asb i. We present an algorithm for computing order-degree sequences and study their impact on independence numbers and connectivities. For instance, if n>=3, then the connectivity of G is at least (smax+smin-n+3), where smax and smin are the largest and the smallest component of the order-degree sequence of G.Christoph Bold