Embeddability Of Graphs Research Articles

Upper embeddability of graphs and products of transpositions associated with edges

Upper embeddability of graphs and products of transpositions associated with edges

Excluded minors for the Klein bottle I. Low connectivity case

Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I we consider the structure of graphs with a 2-vertex-cut that are critical with respect to the Euler genus. A general theorem describing the building blocks is presented. These constituents, called hoppers and cascades, are classified for the case when Euler genus is small. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained. This is the first complete result about obstructions for embeddability of graphs in the Klein bottle, and the outcome is somewhat surprising in the sense that there are considerably fewer excluded minors than expected.

Applications of ordinary voltage graph theory to graph embeddability

AbstractLet p be a prime greater than 5. We show that, while the generalized Petersen graphs of the form have cellular toroidal embeddings, they have no such embeddings having the additional property that a free action of a group on the graph extends to a cellular automorphism of the torus. Such an embedding is called a derived embedding. We also show that does have a derived embedding in the torus, and we show that for any odd q, each generalized Petersen graph of the form has a derived embedding in the Klein bottle, which has the same Euler characteristic as the torus. We close with some comments that frame these results in the light of Abrams and Slilaty's recent work on graphs featuring group actions that extend to spherical embeddings of those graphs.

Antiparallel d-stable traces and a stronger version of ore problem.

In 2013 a novel self-assembly strategy for polypeptide nanostructure design which could lead to significant developments in biotechnology was presented in Gradišar et al. (Nat Chem Bio 9:362-366, 2013). It was since observed that a polyhedron P can be realized by interlocking pairs of polypeptide chains if its corresponding graph G(P) admits a strong trace. It was since also demonstrated that a similar strategy can also be expanded to self-assembly of designed DNA (Kočar, Nat commun 7:1-8, 2016). In this direction, in the present paper we characterize graphs which admit closed walk which traverses every edge exactly once in each direction and for every vertex v, there is no subset N of its neighbors, with [Formula: see text], such that every time the walk enters v from N, it also exits to a vertex in N. This extends Thomassen's characterization (Thomassen, J Combin Theory Ser B 50:198-207, 1990) for the case [Formula: see text].

On computational complexity of length embeddability of graphs

A graph G is embeddable in Rd if the vertices of G can be assigned to points of Rd in such a way that all pairs of adjacent vertices are at distance 1. We show that verifying embeddability of a given graph in Rd is NP-hard in the case d>2 for all reasonable notions of embeddability.

On the distance and multidistance graph embeddability problem

The notions of the distance and multidistance embeddability of a graph in ℝd are natural generalizations of the notion of a distance graph in ℝd. Results on the complexity of the computational problem of verifying distance graph embeddability are briefly surveyed, and new progress in solving this and related problems is described.

Beyond triangle inequality

Knowing accurate positions of nodes in wireless ad hoc and sensor networks is essential for a wide range of pervasive and mobile applications. However, errors are inevitable in distance measurements and we observe that a small number of outliers can degrade localization accuracy drastically. To deal with noisy and outlier ranging results, triangle inequality, is often employed in existing approaches. Our study shows that triangle inequality has many limitations, which make it far from accurate and reliable. In this study, we formally define the outlier detection problem for network localization and build a theoretical foundation to identify outliers based on graph embeddability and rigidity theory. Our analysis shows that the redundancy of distance measurements plays an important role. We then design a bilateration generic cycles-based outlier detection algorithm, and examine its effectiveness and efficiency through a network prototype implementation of MicaZ motes as well as extensive simulations. The results show that our design significantly improves the localization accuracy by wisely rejecting outliers.

A note on the upper embeddability of graphs

In Survey of maximum genus of [J East China Norm Univ Natur Sci, Sep. 2010, No. 5, 1-13], Ren and Li reviewed research developments on maximum genus of graphs in graph embedding theory since 1971, and presented the following two conjectures: Conjecture 1. Let G be a simple connected graph such that each edge is contained in a triangle K 3 . Then G is upper-embeddable. Conjecture 2. Let c be an arbitrary positive number. Then, there exists a natural number N ( c ) such that for every graph G of order n ≥ N ( c ) and minimum degree δ ( G ) ≥ cn , G is upper-embeddable. In this paper, we negate the above two conjectures, and discuss the condition for which the above conjectures is true and obtain some new results. Finally, we present several research problems that would be developed in the future.

Surface embeddability of graphs via homology

This paper presents a characterization of the embeddability on a surface of genus arbitrarily given for a graph. Its specific case for the surface of genus zero leads to the famous planarity theorems given independently by Whitney via duality, MacLane via cycle basis and Lefschetz via double covering at a time.

A suffcient condition on upper embeddability of graphs

In this paper we mainly prove that let G be a (k + 1)-edge-connected simple graph of order n with girth g. Then G is upper embeddable if \( \sum\limits_{v \in I(G)} {d_G (v) \geqslant n - 2g + 5 - 3k - (g - 5)k^2 } \) for any independent set I(G) = {υi | 1 ⩽ i ⩽ k2 + 2}, k = 0,1,2 and the lower bound is tight.

Boolean approaches to graph embeddings related to VLSI

This paper discusses the development of Boolean methods in some topics on graph embeddings which are related to VLSI. They are mainly the general theory of graph embeddability, the orientabilities of a graph and the rectilinear layout of an electronic circuit.

A characterization of embeddability of graphs on surfaces

In this article the Cn-graphs are introduced, by which a characterization of the embeddability of a graph on either an orientable surface or a non-orientable surface is provided.

Obstructions for the Disk and the Cylinder Embedding Extension Problems

LetSbe a closed surface with boundary ∂Sand letGbe a graph. LetK⊆Gbe a subgraph embedded inSsuch that ∂S⊆K. Anembedding extensionofKtoGis an embedding ofGinSthat coincides onKwith the given embedding ofK. Minimal obstructions for the existence of embedding extensions are classified in cases whenSis the disk or the cylinder. Linear time algorithms are presented that either find an embedding extension, or return an obstruction to the existence of extensions. These results are to be used as the corner stones in the design of linear time algorithms for the embeddability of graphs in an arbitrary surface and for solving more general embedding extension problems.

Theoretical results on at most 1-bend embeddability of graphs

1. Introduction Let κ be a non-negative integer. A κ-bend graph is a plane graph in which every edgeis a broken line consisting of at most κ+ 1 horizontal or vertical segments. A bend is any point which is the intersection of a horizontal segment and a verticalsegment of an edge. A planar graph G is κ-rectilinear if it admits a plane embedding Gwhich is a κ-bend graph. In this case G is said to be a κ-embedding of G.

Kuratowski-type theorems do not extend to pseudosurfaces

We show that, in general, graph embeddability in pseudosurfaces cannot be characterized by means of a finite set of forbidden (=irreducible) graphs.

Bidirectional retracting-free double tracings and upper embeddability of graphs

We characterize those finite connected graphs which admit a closed walk such that each edge is traversed once in each direction and such that no edge is succeeded by the same edge in the opposite direction. This solves an old problem of O. Ore.

Matroids Determine the Embeddability of Graphs in Surfaces

The embeddability of a graph in a given surface is determined entirely by the polygon matroid of the graph. That is also true for cellular embeddability in nonorientable surfaces but not in orientable surfaces.

Matroids determine the embeddability of graphs in surfaces

The embeddability of a graph in a given surface is determined entirely by the polygon matroid of the graph. That is also true for cellular embeddability in nonorientable surfaces but not in orientable surfaces.

Isometric subgraphs of Hamming graphs and d-convexity

In this study, we provide criteria of isometric embeddability of graphs in Hamming graphs. We consider ordinary connected graphs with a finite vertex set endowed with the natural metric d(x, y), equal to the number of edges in the shortest chain between the vertices x and y.

Complexity of graph embeddability problems

There exist very efficient algorithms to decide whether a graph is planar. How difficult can it be to decide whether a graph is embeddable in some specified subset S of the plane? It is shown that the time complexity of such embeddability problems can be prescribed in the following sense. To every class C of graphs one can associate a path-connected subset p( C) of the plane, defined as a topological realization of an appropriate countable graph, in such a way that the problem of recognizing the class C is polynomial time equivalent to the problem of testing embeddability in p( C). Incidentally, a similar result is proved on combinatorial embeddability in infinite graphs.