Results 1  10
of
12
Embedding graphs containing K5subdivisions
 Ars Combinatoria
"... Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality al ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Given a nonplanar graph G with a subdivision of K5 as a subgraph, we can either transform the K5subdivision into a K3,3subdivision if it is possible, or else we obtain a partition of the vertices of G\K5 into equivalence classes. As a result, we can reduce a projective planarity or toroidality algorithm to a small constant number of simple planarity checks [6] or to a K3,3subdivision in the graph G. It significantly simplifies algorithms presented in [7], [10] and [12]. We then need to consider only the embeddings on the given surface of a K3,3subdivision, which are much less numerous than those of K5. 1.
CLASSIFICATION OF RINGS WITH GENUS ONE ZERODIVISOR GRAPHS
"... Abstract. This paper investigates properties of the zerodivisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zerodivisor graph has genus 1. 1. ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
Abstract. This paper investigates properties of the zerodivisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zerodivisor graph has genus 1. 1.
The obstructions for toroidal graphs with no K3,3’s
, 2005
"... Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Forbidden minors and subdivisions for toroidal graphs are numerous. We consider the toroidal graphs with no K3,3subdivisions that coincide with the toroidal graphs with no K3,3minors. These graphs admit a unique decomposition into planar components and have short lists of obstructions. We provide the complete lists of four forbidden minors and eleven forbidden subdivisions for the toroidal graphs with no K3,3’s and prove that the lists are sufficient.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...
Forbidden Minors to Graphs with Small Feedback Sets
 DISCRETE MATHEMATICS
, 1996
"... Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper wecharacterize several families of graphs with small feedback sets, namely k 1 Feedback Vertex Set, k 2 Feedback Edge Set and #k 1 ,k 2 ##Feedback Ver ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper wecharacterize several families of graphs with small feedback sets, namely k 1 Feedback Vertex Set, k 2 Feedback Edge Set and #k 1 ,k 2 ##Feedback Vertex#Edge Set, for small integer parameters k 1 and k 2 . Our constructive methods can compute obstruction sets for any minorclosed family of graphs, provided the pathwidth #or treewidth# of the largest obstruction is known.
An algorithm for embedding graphs in the torus
"... An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applyi ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
An efficient algorithm for embedding graphs in the torus is presented. Given a graph G, the algorithm either returns an embedding of G in the torus or a subgraph of G which is a subdivision of a minimal nontoroidal graph. The algorithm based on [13] avoids the most complicated step of [13] by applying a recent result of Fiedler, Huneke, Richter, and Robertson [5] about the genus of graphs in the projective plane, and simplifies other steps on the expense of losing linear time complexity. 1
A Faster Algorithm for Torus Embedding
, 2004
"... Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and di ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Although theoretically practical algorithms for torus embedding exist, they have not yet been successfully implemented and their complexity may be prohibitive to their practicality. We describe a simple exponential algorithm for embedding graphs on the torus (a surface shaped like a doughnut) and discuss how it was inspired by the quadratic time planar embedding algorithm of Demoucron, Malgrange and Pertuiset. We show that it is faster in practice than the only fully implemented alternative (also exponential) and explain how both the algorithm itself and the knowledge gained during its development might be used to solve the wellstudied problem of finding the complete set of torus obstructions.
dx.doi.org/10.2140/pjm.2014.268.371 NONPLANARITY OF UNIT GRAPHS AND CLASSIFICATION OF THE TOROIDAL ONES
"... The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings wi ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
The unit graph of a ring R with nonzero identity is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we derive several necessary conditions for the nonplanarity of the unit graphs of finite commutative rings with nonzero identity, and determine, up to isomorphism, all finite commutative rings with nonzero identity whose unit graphs are toroidal. 1.
Reliability of Planar Multigraphs
, 1996
"... We consider allterminal reliability, one of the more popular models of network reliability. We show that if you consider planar graphs on seven vertices and fourteen edges, the most reliable graph when edges are very reliable (as occurs in most practical applications) is a multigraph. This is the f ..."
Abstract
 Add to MetaCart
(Show Context)
We consider allterminal reliability, one of the more popular models of network reliability. We show that if you consider planar graphs on seven vertices and fourteen edges, the most reliable graph when edges are very reliable (as occurs in most practical applications) is a multigraph. This is the first example known where a multigraph is preferred over all simple graphs with the same order and size. 1 Network Reliability We model a network by an undirected probabilistic graph. For the allterminal model, edges operate independently at random with probability p and the network is operational if the underlying probabilistic graph is connected. Colbourn's monograph is an excellent starting point for a researcher interested in the combinatorics of network reliability [2]. The reliability of a graph G, Rel(G), is the probability that G is connected. A standard formula for the reliability is: Rel(G) = m X i=0 N i (G) \Delta p i \Delta (1 \Gamma p) m\Gammai ; (1) Supported by NSE...