Related papers: Vertex Collapsing and Cut Ideals

Bipartite graphs model the relationship between two disjoint sets of objects. They have a wide range of applications and are often visualized as a 2-layered drawing, where each set of objects is visualized as a set of vertices (points) on…

In this paper, we introduce the concept of complementary edge ideals of graphs and study their algebraic properties and invariants.

We develop a theory of minors for alternating dimaps --- orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that…

The widely studied edge modification problems ask how to minimally alter a graph to satisfy certain structural properties. In this paper, we introduce and study a new edge modification problem centered around transforming a given graph into…

Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower…

It is known that the alternation hierarchy of least and greatest fixpoint operators in the mu-calculus is strict. However, the strictness of the alternation hierarchy does not necessarily carry over when considering restricted classes of…

We consider the classical minimum and maximum cut problems: find a partition of vertices of a graph into two disjoint subsets that minimize or maximize the sum of the weights of edges with endpoints in different subsets. It is known that if…

Modern methods of graph theory describe a graph up to isomorphism, which makes it difficult to create mathematical models for visualizing graph drawings on a plane. The topological drawing of the planar part of a graph allows representing…

In communication field, an important issue is to group users and base stations to as many as possible subnetworks satisfying certain interference constraints. These problems are usually formulated as a graph partition problems which…

What scaling limits govern neural network training dynamics when model size and training time grow in tandem? We show that despite the complex interactions between architecture, training algorithms, and data, compute-optimally trained…

In graph modification problems, one is given a graph G and the goal is to apply a minimum number of modification operations (such as edge deletions) to G such that the resulting graph fulfills a certain property. For example, the Cluster…

The `lifting` or `splitting-off` operation on graphs is performed by deleting two edges sv and sw having a common end s and adding a new edge between v and w. Such a lift is considered good if it preserves a certain local edge-connectivity…

We consider changes in properties of a subgraph of an infinite graph resulting from the addition of open edges of Bernoulli percolation on the infinite graph to the subgraph. We give the triplet of an infinite graph, one of its subgraphs,…

The empirical emergence of neural collapse -- a surprising symmetry in the feature representations of the training data in the penultimate layer of deep neural networks -- has spurred a line of theoretical research aimed at its…

We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive…

Graph vertex ordering is widely employed in spatial data analysis, especially in urban analytics, where street graphs serve as spatial discretization for modeling and simulation. It is also crucial for visualization, as many methods require…

We study a random graph model in continuous time. Each vertex is partially copied with the same rate, i.e.\ an existing vertex is copied and every edge leading to the copied vertex is copied with independent probability $p$. In addition,…

In this paper, we study rooted products of graphs from the perspective of combinatorial commutative algebra. For edge ideals, we introduce the 2-Cohen-Macaulayness with respect to a vertex and use it to investigate when edge ideals of…

The cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian…

Spectral graph theory is well known and widely used in computer vision. In this paper, we analyze image segmentation algorithms that are based on spectral graph theory, e.g., normalized cut, and show that there is a natural connection…