Results

**11 - 18**of**18**### 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 ..."

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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

### 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 ..."

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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

### Non-separating planar chains in 4-connected graphs

, 2002

"... In this paper, we describe an O(|V (G) | 2) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed ..."

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In this paper, we describe an O(|V (G) | 2) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed networks, as well as the study of non-separating induced paths in highly connected graphs.

### 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 ..."

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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

### Chain decompositions of 4-connected graphs

, 2003

"... In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an O(|V (G) | 2 |E(G)|) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be imp ..."

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In this paper we give a decomposition of a 4-connected graph G into nonseparating chains, which is similar to an ear decomposition of a 2-connected graph. We also give an O(|V (G) | 2 |E(G)|) algorithm that constructs such a decomposition. In applications, the asymptotic performance can often be improved to O(|V (G) | 3). This decomposition will be used to find four independent spanning trees in a 4connected graph.

### Simplified O(n) Planarity Algorithms

, 2001

"... A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This pape ..."

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A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Due to the wealth of interest from the computer science community, there are a number of remarkable but complex O(n) planar embedding algorithms. This paper presents an O(n) planar embedding algorithm that avoids a number of the complexities of prior approaches (an early version of this work was presented at the January 1999 Symposium on Discrete Algorithms). In July 1999

### Correcting and Implementing the PC-tree Planarity Algorithm

"... A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Recent research eorts have produced new algorithms for solving planarity-related problems. Shih and Hsu proposed a linear-time algorithm based on a data st ..."

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A graph is planar if it can be drawn on the plane with vertices at unique locations and no edge intersections except at the vertex endpoints. Recent research eorts have produced new algorithms for solving planarity-related problems. Shih and Hsu proposed a linear-time algorithm based on a data structure they named PC-tree, which is similar to but much simpler than a PQ-tree. However, their presentation does not explain in detail how to implement the routines that manipulate a PC-tree, and there are some nontrivial correctness and run-time issues that were not addressed in their paper. So it is far from trivial to derive a proper linear-time implementation from their description. This paper presents additions to the theoretical framework of the PC-tree algorithm that are necessary to achieve correctness and linear running time. A linear-time implementation that addresses the issues raised in this paper was developed in the LEDA platform and is available.