Topic
Circle graph
About: Circle graph is a research topic. Over the lifetime, 602 publications have been published within this topic receiving 15275 citations.
Papers published on a yearly basis
Papers
Book•
1 Jan 1980
TL;DR: This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems and remains a stepping stone from which the reader may embark on one of many fascinating research trails.
Abstract: Algorithmic Graph Theory and Perfect Graphs, first published in 1980, has become the classic introduction to the field. This new Annals edition continues to convey the message that intersection graph models are a necessary and important tool for solving real-world problems. It remains a stepping stone from which the reader may embark on one of many fascinating research trails. The past twenty years have been an amazingly fruitful period of research in algorithmic graph theory and structured families of graphs. Especially important have been the theory and applications of new intersection graph models such as generalizations of permutation graphs and interval graphs. These have lead to new families of perfect graphs and many algorithmic results. These are surveyed in the new Epilogue chapter in this second edition. New edition of the "Classic" book on the topic Wonderful introduction to a rich research area Leading author in the field of algorithmic graph theory Beautifully written for the new mathematician or computer scientist Comprehensive treatment
4,370 citations
TL;DR: In this article, the problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, 1)-matrices: when can the rows of such a matrix be permuted so as to make the 1's in each colum appear consecutively.
Abstract: : According to present genetic theory, the fine structure of genes consists of linearly ordered elements. A mutant gene is obtained by alteration of some connected portion of this structure. By examining data obtained from suitable experiments, it can be determined whether or not the blemished portions of two mutant genes intersect or not, and thus intersection data for a large number of mutants can be represented as an undirected graph. If this graph is an interval graph, then the observed data is consistent with a linear model of the gene. The problem of determining when a graph is an interval graph is a special case of the following problem concerning (0, 1)-matrices: When can the rows of such a matrix be permuted so as to make the 1's in each colum appear consecutively. A complete theory is obtained for this latter problem, culminating in a decomposition theorem which leads to a rapid algorithm for deciding the question, and for constructing the desired permutation when one exists.
1,488 citations
TL;DR: This paper shows that the cliques of the intersection graph provide a first set of facets for the polyhedron in question, and it is shown that the cycles without chords of odd length of the intersections graph give rise to a further set of facet.
Abstract: In this paper we address ourselves to identifying facets of the set packing polyhedron, i.e., of the convex hull of integer solutions to the set covering problem with equality constraints and/or constraints of the form "ź". This is done by using the equivalent node-packing problem derived from the intersection graph associated with the problem under consideration. First, we show that the cliques of the intersection graph provide a first set of facets for the polyhedron in question. Second, it is shown that the cycles without chords of odd length of the intersection graph give rise to a further set of facets. A rather strong geometric property of this set of facets is exhibited.
665 citations
TL;DR: The word problem for products of symmetric groups, the circular arc graph coloring problem, and the circle graph coloring Problem are proved to be $NP$-complete and the problem of determining whether a given circular arcs graph is K-colorable is shown to be solvable in polynomial time.
Abstract: The word problem for products of symmetric groups, the circular arc graph coloring problem, and the circle graph coloring problem, as well as several related problems, are proved to be $NP$-complete. For any fixed number K of colors, the problem of determining whether a given circular arc graph is K-colorable is shown to be solvable in polynomial time.
473 citations
TL;DR: Efficient algorithms for finding a maximum clique and a maximum independent set of circle graphs that require at most n3 steps, where n is the number of vertices in the graph.
Abstract: Consider a family of chords in a circle. A circle graph is obtained by representing each chord by a vertex, two vertices being connected by an edge when the corresponding chords intersect. In this paper, we describe efficient algorithms for finding a maximum clique and a maximum independent set of circle graphs. These algorithms require at most n3 steps, where n is the number of vertices in the graph.
205 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 1 |
| 2022 | 4 |
| 2021 | 16 |
| 2020 | 9 |
| 2019 | 17 |
| 2018 | 10 |