Related papers: Enumerating Minimal Defensive Alliances

The question to enumerate all inclusion-minimal connected dominating sets in a graph of order $n$ in time significantly less than $2^n$ is an open question that was asked in many places. We answer this question affirmatively, by providing…

Let $\G(V,E)$ be a simple graph without loops nor multiple edges. A nonempty subset $S \subseteq V$ is said a {\em global offensive alliance} if every vertex $v \in V- S$ satisfies that $\d_S(v) \geq \d_{\overline{S}}(v)+1$. The {\em global…

We consider the problem of finding decentralized strategies for multi-agent perimeter defense games. In this work, we design a graph neural network-based learning framework to learn a mapping from defenders' local perceptions and the…

In this paper, we prove an inequality on the cardinalities of the minimum size global defensive alliance and the minimum size global offensive alliance. A global defensive alliance is a dominating set such that when any point inside a…

We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of…

Deep neural networks (DNNs) have been widely applied to various applications, including image classification, text generation, audio recognition, and graph data analysis. However, recent studies have shown that DNNs are vulnerable to…

The vulnerability of machine learning models to adversarial attacks has been attracting considerable attention in recent years. Most existing studies focus on the behavior of stand-alone single-agent learners. In comparison, this work…

Graph-based classification methods are widely used for security and privacy analytics. Roughly speaking, graph-based classification methods include collective classification and graph neural network. Evading a graph-based classification…

The alliance polynomial of a graph $\Gamma$ with order $n$ and maximum degree $\delta_1$ is the polynomial $A(\Gamma; x) = \sum_{k=-\delta_1}^{\delta_1} A_{k}(\Gamma) \, x^{n+k}$, where $A_{k}(\Gamma)$ is the number of exact defensive…

We investigate the vulnerabilities of consensus-based distributed optimization protocols to nodes that deviate from the prescribed update rule (e.g., due to failures or adversarial attacks). We first characterize certain fundamental…

Graph classification is crucial in network analyses. Networks face potential security threats, such as adversarial attacks. Some defense methods may trade off the algorithm complexity for robustness, such as adversarial training, whereas…

We investigate the relationship between global offensive $k$-alliances and some characteristic sets of a graph including $r$-dependent sets and $\tau$-dominating sets. As a consequence of the study, we obtain bounds on the global offensive…

Adversarial attacks to graph analytics are gaining increased attention. To date, two lines of countermeasures have been proposed to resist various graph adversarial attacks from the perspectives of either graph per se or graph neural…

In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and…

Let $G=(V,E)$ be a simple graph. For a nonempty set $X\subset V,$ and a vertex $v\in V,$ $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X.$ A nonempty set $S\subset V$ is an \emph{offensive $r$-alliance} in $G$ if…

An offensive alliance in a graph $\Gamma=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$'s closed neighborhood are in $S$. In the case of strong offensive…

For a graph $G=(V,E)$, a set $S\subseteq V$ is a dominating set if every vertex in $V-S$ has at least a neighbor in $S$. A dominating set $S$ is a global offensive alliance if for each vertex $v$ in $V-S$ at least half the vertices from the…

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient…

Graph Neural Networks (GNNs) are increasingly important given their popularity and the diversity of applications. Yet, existing studies of their vulnerability to adversarial attacks rely on relatively small graphs. We address this gap and…

We develop a functional analytic approach for the study of nonlocal minimal graphs. Through this, we establish existence and uniqueness results, a priori estimates, comparison principles, rearrangement inequalities, and the equivalence of…