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Cascading failures are one of the main reasons for blackouts in electrical power grids. Stable power supply requires a robust design of the power grid topology. Currently, the impact of the grid structure on the grid robustness is mainly…
A graph with n vertices is 1-planar if it can be drawn in the plane such that each edge is crossed at most once, and is optimal if it has the maximum of 4n-8 edges. We show that optimal 1-planar graphs can be recognized in linear time. Our…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
This article introduces and studies a new class of graphs motivated by discrete curvature. We call a graph resistance nonnegative if there exists a distribution on its spanning trees such that every vertex has expected degree at most two in…
An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code…
Graph Attention Networks(GATs) are useful deep learning models to deal with the graph data. However, recent works show that the classical GAT is vulnerable to adversarial attacks. It degrades dramatically with slight perturbations.…
There has been recent growing interest in graph theoretical properties known as r- and (r,s)-robustness. These properties serve as sufficient conditions guaranteeing the success of certain consensus algorithms in networks with misbehaving…
Recent studies have shown that attackers can catastrophically reduce the performance of GNNs by maliciously modifying the graph structure or node features on the graph. Adversarial training, which has been shown to be one of the most…
Supervised learning on graphs is a challenging task due to the high dimensionality and inherent structural dependencies in the data, where each edge depends on a pair of vertices. Existing conventional methods are designed for standard…
This work introduces a novel algorithm for finding the connected components of a graph where the vertices and edges are grouped into sets defining a Set--Based Graph. The algorithm, under certain restrictions on those sets, has the…
Graph Neural Networks (GNNs) have shown their great ability in modeling graph structured data. However, real-world graphs usually contain structure noises and have limited labeled nodes. The performance of GNNs would drop significantly when…
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in…
In this paper, we present a probabilistic analysis of iterative node-based verification-based (NB-VB) recovery algorithms over irregular graphs in the context of compressed sensing. Verification-based algorithms are particularly interesting…
We study the richness of the ensemble of graphical structures (i.e., unlabeled graphs) of the one-dimensional random geometric graph model defined by $n$ nodes randomly scattered in $[0,1]$ that connect if they are within the connection…
Circular layouts are a popular graph drawing style, where vertices are placed on a circle and edges are drawn as straight chords. Crossing minimization in circular layouts is \NP-hard. One way to allow for fewer crossings in practice are…
Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique…
In this letter, we prove that under mild conditions, the scaled graph of a reset control system is bounded by the scaled graph of its underlying base linear system, i.e., the system without resets. Building on this new insight, we establish…
In this paper we study local routing strategies on geometric graphs. Such strategies use geometric properties of the graph like the coordinates of the current and target nodes to route. Specifically, we study routing strategies in the…
The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a…
A graph neural network transforms features in each vertex's neighborhood into a vector representation of the vertex. Afterward, each vertex's representation is used independently for predicting its label. This standard pipeline implicitly…