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The concept of Roman domination has been a subject of intrigue for more than two decades with the fundamental Roman domination problem standing out as one of the most significant challenges in this field. This article studies a practically…
In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent…
The Roman domination in graphs is well-studied in graph theory. The topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. The deployment of the legions around the…
$k$-defensive domination, a variant of the classical domination problem on graphs, seeks a minimum cardinality vertex set providing a surjective defense against any attack on vertices of cardinality bounded by a parameter $k$. The problem…
Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman…
The Roamn domination problem is one important combinatorial optimization problem that is derived from an old story of defending the Roman Empire and now regains new significance in cyber space security, considering backups in the face of a…
The study of Roman domination has evolved to encompass a variety of challenging extensions, each contributing to the broader understanding of domination problems in graph theory. This paper explores the Perfect Location Signed Roman…
The problem Defensive $\delta$-Covering, for some covering range $\delta > 0$, is a continuous facility location problem on undirected graphs where all edges have unit length. It is a generalization of Defensive Dominating Set and…
Motivated by resource defense models in networks, such as protecting territories with varying legion strengths, let $k \geq 2$ be an integer. Roman $k$-domination and strong Roman $k$-domination generalize Roman, double Roman, Italian, and…
The signed double Roman domination problem is a combinatorial optimization problem on a graph asking to assign a label from $\{\pm{}1,2,3\}$ to each vertex feasibly, such that the total sum of assigned labels is minimized. Here feasibility…
Roman domination and its higher-order extensions have attracted considerable attention due to their natural interpretation in terms of defensive resource allocation on networks. The recently introduced $[k]$-Roman domination framework…
Graph neural networks (GNNs) have attracted increasing interests. With broad deployments of GNNs in real-world applications, there is an urgent need for understanding the robustness of GNNs under adversarial attacks, especially in realistic…
Intercepting a criminal using limited police resources presents a significant challenge in dynamic crime environments, where the criminal's location continuously changes over time. The complexity is further heightened by the vastness of the…
In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of…
Graph convolutional networks (GCNs) have been shown to be vulnerable to small adversarial perturbations, which becomes a severe threat and largely limits their applications in security-critical scenarios. To mitigate such a threat,…
We study the m-Eternal Domination problem, which is the following two-player game between a defender and an attacker on a graph: initially, the defender positions k guards on vertices of the graph; the game then proceeds in turns between…
We consider a dynamic model for competition in a social network, where two strategic agents have fixed beliefs and the non-strategic/regular agents adjust their states according to a distributed consensus protocol. We suppose that one…
The domination problem and its variants represent a classical domain within algorithmic graph theory. Among these variants, the paired-domination problem holds particular prominence due to its real-world implications in security and…
Given a positive integer $k$, a $k$-dominating set in a graph $G$ is a set of vertices such that every vertex not in the set has at least $k$ neighbors in the set. A total $k$-dominating set, also known as a $k$-tuple total dominating set,…
Given a graph $G=(V,E)$, the dominating number of a graph is the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating function of $G$ is…