Results

**11 - 17**of**17**### Untangling Tanglegrams: Comparing Trees by their Drawings ∗

, 2009

"... A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimi ..."

Abstract
- Add to MetaCart

A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimizations problems in tanglegram drawings. We show a linear time algorithm to decide if a tanglegram admits a planar embedding by a reduction to the planar graph drawing problem. This problem was also studied by Fernau, Kauffman and Poths (FSTTCS 2005). A similar reduction to a graph crossing problem also helps to solve an open problem they posed, showing a fixed-parameter tractable algorithm for minimizing the number of crossings over all d-ary trees. For the case where one tree is fixed, we show an O(n log n) algorithm to determine the drawing of the second tree that minimizes the number of crossings. This improves the bound from earlier methods. We introduce a new optimization criterion using Spearman’s footrule distance and give an O(n 2) algorithm. We also show integer programming formulations to quickly obtain tanglegram drawings that minimize the two optimization measures discussed. We prove lower bounds on the maximum gap between the optimal solution and the heuristic of Dwyer and Schreiber (Austral. Symp. on Info. Vis. 2004) to minimize crossings. 1

### Minimizing the Number of Label Transitions Around a Nonseparating Vertex of a Planar

, 2012

"... We study the minimum number of label transitions around a given vertex v0 in a planar multigraph G, in which the edges incident with v0 are labelled with integers 1,..., l, and the minimum is taken over all embeddings of G in the plane. For a fixed number of labels, a lineartime fixed-parameter trac ..."

Abstract
- Add to MetaCart

We study the minimum number of label transitions around a given vertex v0 in a planar multigraph G, in which the edges incident with v0 are labelled with integers 1,..., l, and the minimum is taken over all embeddings of G in the plane. For a fixed number of labels, a lineartime fixed-parameter tractable algorithm that computes the minimum number of label transitions around v0 is presented. If the number of labels is unconstrained, then the problem of deciding whether the minimum number of label transitions is at most k is NP-complete. Submitted:

### Subgraph Homeomorphism via the Edge Addition Planarity Algorithm

, 2012

"... This paper extends the edge addition planarity algorithm from Boyer and Myrvold to provide a new way of solving the subgraph homeomorphism problem for K2,3, K4, and K3,3. These extensions derive much of their behavior and correctness from the edge addition planarity algorithm, providing an alternati ..."

Abstract
- Add to MetaCart

This paper extends the edge addition planarity algorithm from Boyer and Myrvold to provide a new way of solving the subgraph homeomorphism problem for K2,3, K4, and K3,3. These extensions derive much of their behavior and correctness from the edge addition planarity algorithm, providing an alternative perspective on these subgraph homeomorphism problems based on affinity with planarity rather than triconnectivity. Reference implementations of these algorithms have been made available in an open source project

### An Algorithm for 3D-biplanar Graph Drawing

"... We introduce the concept of 3D-biplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NP-complete and present a randomized parameterized algorith ..."

Abstract
- Add to MetaCart

We introduce the concept of 3D-biplanar drawing in which we partition a graph into two planar induced subgraphs. Our goal is to find such a partition with the minimum number of edges between the two partitions. We prove that this problem is NP-complete and present a randomized parameterized algorithm with O(n k) time, where k is the ratio of the optimal solution to the min-cut size of the graph. 1