Related papers: On robust toric ideals of weighted oriented graphs

In this paper, we study the large-scale structure of dense regular graphs. This involves the notion of robust expansion, a recent concept which has already been used successfully to settle several longstanding problems. Roughly speaking, a…

In this paper, we study toric ideals associated with multichains of posets. It is shown that the comparability graph of a poset is chordal if and only if there exists a quadratic Gr\"obner basis of the toric ideal of the poset. Strong…

One can associate to a bipartite graph a so-called edge ring whose spectrum is an affine normal toric variety. We characterize the faces of the (edge) cone associated to this toric variety in terms of some independent sets of the bipartite…

Let $\mathcal{D}$ be a weighted oriented graph and let $I(\mathcal{D})$ be its edge ideal in a polynomial ring $R$. We give the formula of Castelnuovo-Mumford regularity of $R/I(\mathcal{D})$ when $\mathcal{D}$ is a weighted oriented path…

A graph $G$ is a $D\!D_2$-graph if it has a pair $(D,D_2)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $D_2$ is a 2-dominating set of $G$. We provide several characterizations and hardness results concerning…

We show that any graph that is generically globally rigid in $\mathbb{R}^d$ has a realization in $\mathbb{R}^d$ that is both generic and universally rigid. This also implies that the graph also must have a realization in $\mathbb{R}^d$ that…

This paper studies the problem of designing networks that are strong structurally controllable, and robust simultaneously. For given network specifications, including the number of nodes $N$, the number of leaders $N_L$, and diameter $D$,…

Two directed graphs are called covariance equivalent if they induce the same set of covariance matrices, up to a Lebesgue measure zero set, on the random variables of their associated linear structural equation models. For acyclic graphs,…

Let ${\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\cal G})$ to be the…

We examine the generic local and global rigidity of various graphs in R^d. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity and hypothesized that they…

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an…

Perfect graphs form one of the distinguished classes of finite simple graphs. In 2006, Chudnovsky, Robertson, Seymour and Thomas proved that a graph is perfect if and only if it has no odd holes and no odd antiholes as induced subgraphs,…

Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gr\"obner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely…

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals of vertex-weighted oriented unicyclic graphs. These formulas are in function of the weight of the vertices, the numbers of edges. We…

Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral…

We investigate certain structural properties of random interdependent networks. We start by studying a property known as $r$-robustness, which is a strong indicator of the ability of a network to tolerate structural perturbations and…

We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: (1) A toroidal graph embedding $\Gamma$ is positive equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors…

A bar-joint framework $(G,p)$ in the Euclidean space $\mathbb{E}^d$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ in $\mathbb{E}^d$ with the edge lengths of $(G,p)$. Building on key results of…

A $d$-dimensional (bar-and-joint) framework $(G,p)$ consists of a graph $G=(V,E)$ and a realisation $p:V\to \mathbb{R}^d$. It is rigid if every continuous motion of the vertices which preserves the lengths of the edges is induced by an…

Deep learning models have been shown to be vulnerable to adversarial attacks. This perception led to analyzing deep learning models not only from the perspective of their performance measures but also their robustness to certain types of…