Invariant Edges Research Articles

$b$-Invariant Edges in Essentially 4-Edge-Connected Near-Bipartite Cubic Bricks

A brick is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is $b$-invariant if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is essentially 4-edge-connected if it does not contain nontrivial 3-cuts. A brick $G$ is near-bipartite if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered.
 Kothari, de Carvalho, Lucchesi and Little proved that each essentially 4-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges. Moreover, they made a conjecture: every essentially 4-edge-connected cubic near-bipartite brick $G$, distinct from $K_4$, has at least $|V(G)|/2$ $b$-invariant edges. We confirm the conjecture in this paper. Furthermore, all the essentially 4-edge-connected cubic near-bipartite bricks, the numbers of $b$-invariant edges of which attain the lower bound, are presented.

On Essentially 4-Edge-Connected Cubic Bricks

Lovász (1987) proved that every matching covered graph $G$ may be uniquely decomposed into a list of bricks (nonbipartite) and braces (bipartite); we let $b(G)$ denote the number of bricks. An edge $e$ is removable if $G-e$ is also matching covered; furthermore, $e$ is $b$-invariant if $b(G-e)=1$, and $e$ is quasi-$b$-invariant if $b(G-e)=2$. (Each edge of the Petersen graph is quasi-$b$-invariant.)
 A brick $G$ is near-bipartite if it has a pair of edges $\{e,f\}$ so that $G-e-f$ is matching covered and bipartite; such a pair $\{e,f\}$ is a removable doubleton. (Each of $K_4$ and the triangular prism $\overline{C_6}$ has three removable doubletons.) Carvalho, Lucchesi and Murty (2002) proved a conjecture of Lovász which states that every brick, distinct from $K_4$, $\overline{C_6}$ and the Petersen graph, has a $b$-invariant edge.
 A cubic graph is essentially $4$-edge-connected if it is $2$-edge-connected and if its only $3$-cuts are the trivial ones; it is well-known that each such graph is either a brick or a brace; we provide a graph-theoretical proof of this fact.
 We prove that if $G$ is any essentially $4$-edge-connected cubic brick then its edge-set may be partitioned into three (possibly empty) sets: (i) edges that participate in a removable doubleton, (ii) $b$-invariant edges, and (iii) quasi-$b$-invariant edges; our Main Theorem states that if $G$ has two adjacent quasi-$b$-invariant edges, say $e_1$ and $e_2$, then either $G$ is the Petersen graph or the (near-bipartite) Cubeplex graph, or otherwise, each edge of $G$ (distinct from $e_1$ and $e_2$) is $b$-invariant. As a corollary, we deduce that each essentially $4$-edge-connected cubic non-near-bipartite brick $G$, distinct from the Petersen graph, has at least $|V(G)|$ $b$-invariant edges.

Relaxation and edge reconstruction in integer quantum Hall systems

The interplay between the confinement potential and the electron–electron interactions causes reconstructions of quantum Hall edges. We study the consequences of this edge reconstruction for the relaxation of hot electrons injected into integer quantum Hall edge states. In translationally invariant edges, the relaxation of hot electrons is governed by three-body collisions, which are sensitive to the electron dispersion and thus to reconstruction effects. We show that the relaxation rates are significantly altered in different reconstruction scenarios.

Invariant syllable skeleton, complex segments and word edges

San Duanmu'sSyllable Structure: The Limits of Variationraises a number of questions that are of general interest for phonological theory. Of special interest here are: the genesis and management of linearity in complex segments, the place of analogy (or paradigm uniformity) in grammar, the role of morphology in accounting for phonological patterns, the balance of static (distributional patterns) and dynamic (phonological processes) evidence for syllable structure, the role of stress in syllabification, and the import of corpus-based data for phonological analysis. In each case, Duanmu's proposals are evaluated according to their intrinsic consistency, the empirical record and the relevant body of literature. Alternative ways of handling the phenomena are offered, and these are fairly traditional in most cases. Duanmu's book is particularly relevant in the current constitution of the field where the see-saw movement between computation and representations seems to swing back in direction of the latter after having long been immobilised on the computational end. Standing clearly on the representational side, the theory exposed in the book aims to show that all surface strings may be reduced to a fixed and invariant syllable template, C(onsonant)V(owel)X. This enterprise is interesting especially in presence of another representationally-oriented theory, CVCV (Lowenstamm 1996, Scheer 2004), which also aims at reducing surface variation to an invariant syllabic skeleton, made of a monotonic sequence of CV units. However, the CVX and the CVCV templates are quite distinct, and the strategies that are used in order to accommodate the surface string are opposite (shrinking in the former case, expanding in the latter).

Feature Detection by Color Tensor-Based Photometric Invariants

Although the majority of images are recorded in color format nowadays, computer vision research is still mostly restricted to luminance-based feature detection. In this paper, we combine the features based on the color tensor with photometric invariant derivatives to arrive at photometric invariant features. The combination of the photometric invariance theory and tensor based features allows for detection of a variety of features such as photometric invariant edges, corners. Experiments show that the proposed features are robust to scene incidental events and perform well in real-world scene.

Robust photometric invariant features from the color tensor

Luminance-based features are widely used as low-level input for computer vision applications, even when color data is available. The extension of feature detection to the color domain prevents information loss due to isoluminance and allows us to exploit the photometric information. To fully exploit the extra information in the color data, the vector nature of color data has to be taken into account and a sound framework is needed to combine feature and photometric invariance theory. In this paper, we focus on the structure tensor, or color tensor, which adequately handles the vector nature of color images. Further, we combine the features based on the color tensor with photometric invariant derivatives to arrive at photometric invariant features. We circumvent the drawback of unstable photometric invariants by deriving an uncertainty measure to accompany the photometric invariant derivatives. The uncertainty is incorporated in the color tensor, hereby allowing the computation of robust photometric invariant features. The combination of the photometric invariance theory and tensor-based features allows for detection of a variety of features such as photometric invariant edges, corners, optical flow, and curvature. The proposed features are tested for noise characteristics and robustness to photometric changes. Experiments show that the proposed features are robust to scene incidental events and that the proposed uncertainty measure improves the applicability of full invariants.

A Linear Algorithm for Finding the Invariant Edges of an Edge-Weighted Graph

Given an edge-weighted graph where all weights are nonnegative reals, an edge reweighting is an assignment of nonnegative reals to edges such that, for each vertex, the sums of given and new weights assigned to the edges incident on the vertex do coincide. An edge is then said to be invariant if its weight is the same for any edge reweighting. We show that the set of invariant edges of an arbitrary edge-weighted graph can be determined in time linear in the size of the underlying graph. Moreover, an application to the security of statistical data is discussed.

PicToSeek: combining color and shape invariant features for image retrieval

We aim at combining color and shape invariants for indexing and retrieving images. To this end, color models are proposed independent of the object geometry, object pose, and illumination. From these color models, color invariant edges are derived from which shape invariant features are computed. Computational methods are described to combine the color and shape invariants into a unified high-dimensional invariant feature set for discriminatory object retrieval. Experiments have been conducted on a database consisting of 500 images taken from multicolored man-made objects in real world scenes. From the theoretical and experimental results it is concluded that object retrieval based on composite color and shape invariant features provides excellent retrieval accuracy. Object retrieval based on color invariants provides very high retrieval accuracy whereas object retrieval based entirely on shape invariants yields poor discriminative power. Furthermore, the image retrieval scheme is highly robust to partial occlusion, object clutter and a change in the object's pose. Finally, the image retrieval scheme is integrated into the PicToSeek system on-line at http://www.wins.uva.nl/research/isis/PicToSeek/ for searching images on the World Wide Web.

The network flows approach for matrices with given row and column sums

We study the set of integral matrices with given row and column sums where each position has a bound on the size of the entry in it. Such matrices correspond to maximum integral flows in certain networks. The well known existence theorem follows from the max flow-min cut theorem. A general network flow result, specialized to our setting, yields a useful interchange theorem which has a number of corollaries. Prompted by another network flow result, new results are obtained for invariant positions i.e. entries which are the same regardless of which matrix is chosen. This leads to a classification of invariant edges in graphs of a given degree sequence, important in the study of split graphs and threshold graphs.