Simpler, Linear-Time Transitive Orientation via Lexicographic Breadth-First Search

具有多叶或少叶的广度优先搜索树

Breadth-First Search Trees with Many or Few Leaves

研究了在广度优先搜索、通用搜索和字典序广度优先搜索中，寻找具有最大或最小叶数的搜索树问题，分析了其复杂性。

Graph Searches and Their End Vertices

Graph search, the process of visiting vertices in a graph in a specific order, has demonstrated magical powers in many important algorithms. But a systematic study was only initiated by Corneil et al.~a decade ago, and only by then we started to realize how little we understand it. Even the apparently naïve question "which vertex can be the last visited by a graph search algorithm," known as the end vertex problem, turns out to be quite elusive. We give a full picture of all maximum cardinality searches on chordal graphs, which implies a polynomial-time algorithm for the end vertex problem of …

Practical and Efficient Circle Graph Recognition

Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, α(n + m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the au…

A Parallel Algorithm to Test Chordality of Graphs

We present a simple parallel algorithm to test chordality of graphs which is based on the parallel Lexicographical Breadth-First Search algorithm. In total, the algorithm takes time O(N ) on N-threads machine and it performs work O(N 2 ) , where N is the number of vertices in a graph. Our implementation of the algorithm uses a GPU environment Nvidia CUDA C. The algorithm is implemented in CUDA 4.2 and it has been tested on Nvidia GeForce GTX 560 Ti of compute capability 2.1. At the end of the thesis we present the results achieved by our implementation and compare them with the results achieve…

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental ch…

Medians in median graphs and their cube complexes in linear time

The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the $\ell_1$-cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cu…