Protecting a Graph with Mobile Guards

Mobile guards on the vertices of a graph are used to defend it against attacks on either its vertices or its edges. Various models for this problem have been proposed. In this survey we describe a number of these models with particular attention to the case when the attack sequence is infinitely long and the guards must induce some particular configuration before each attack, such as a dominating set or a vertex cover. Results from the literature concerning the number of guards needed to successfully defend a graph in each of these problems are surveyed.Comment: 29 pages, two figures, surve

On the q-criticality of graphs with respect to secure graph domination

Abstract A subset X of the vertex set of a graph G is a secure dominating set of G if each vertex of G which is not in X is adjacent to some vertex in X and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X − {v}) ∪ {u} is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. The notion of secure graph domination finds applications in the generic setting where the vertex set of G represents distributed locations in some spatial domain and the edges of G represent links between these locations. Patrolling guards, each stationed at one of these locations, may move along the links in order to protect the graph. A minimum secure dominating set of G then represents a smallest collection of locations at which guards may be stationed so that the entire location complex modelled by G is protected in the sense that if a security concern arises at location u, there will either be a guard stationed at that location who can deal with the problem, or else a guard dealing with the problem from an adjacent location v will still leave the location complex protected after moving from location v to location u in order to deal with the problem. A graph G is q-critical if the smallest arbitrary subset of edges whose removal from G necessarily increases the secure domination number, has cardinality q. The notion of qcriticality is important in applications such as the one mentioned above, because it provides threshold information as to the number of edge failures (perhaps caused by an adversary) that will necessitate the hiring of additional guards to secure the location complex. Denote by Ωn the largest value of q for which q-critical graphs of order n exist. It has previously been established that Ω2 = 1, Ω3 = 2, Ω4 = 4, Ω5 = 6 and Ω6 = 9. In this paper we present a repository of all q-critical graphs of orders 2, 3, 4, 5 and 6 for all admissible values of q and we also establish the previously unknown values Ω7 = 12, Ω8 = 17 and Ω9 = 23. These values support an existing conjecture that Ωn = n 2 − 2n + 5 for all n ≥ 7

Algorithmic Complexity of Isolate Secure Domination in Graphs

A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set $S\subseteq V$ is an isolate secure dominating set (ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate secure domination number, and is denoted by $\gamma_{0s}(G)$. Given a graph $G=(V,E)$ and a positive integer $k,$ the ISDM problem is to check whether $G$ has an isolate secure dominating set of size at most $k.$ We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.Comment: arXiv admin note: substantial text overlap with arXiv:2002.00002; text overlap with arXiv:2001.1125

Algorithmic complexity of isolate secure domination in graphs

A dominating set S is an Isolate Dominating Set (IDS) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S subset of V is an isolate secure dominating set (ISDS), if for each vertex u is an element of V \ S, there exists a neighboring vertex v of u in S such that (S \ {v}) boolean OR {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by gamma(0s) (G). We give isolate secure domination number of path and cycle graphs. Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.Publisher's Versio

Algorithmic Aspects of Secure Connected Domination in Graphs

Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \}$ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $\gamma_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $G$ and a positive integer $k,$ the Secure Connected Domination (SCDM) problem is to check whether $G$ has a secure connected dominating set of size at most $k.$ In this paper, we prove that the SCDM problem is NP-complete for doubly chordal graphs, a subclass of chordal graphs. We investigate the complexity of this problem for some subclasses of bipartite graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem is to find a secure connected dominating set of minimum size in the input graph. We propose a $(\Delta(G)+1)$ - approximation algorithm for MSCDS, where $\Delta(G)$ is the maximum degree of the input graph $G$ and prove that MSCDS cannot be approximated within $(1 -\epsilon) ln(| V |)$ for any $\epsilon > 0$ unless $NP \subseteq DTIME(| V |^{O(log log | V |)})$ even for bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs with $\Delta(G)=4$

The cost of edge removal in graph domination

CITATION:A. P. de Villiers (2020): The cost of edge removal in graph domination, AKCE International Journal of Graphs and CombinatoricsABSTRACT A vertex set D of a graph G is a dominating set of G if each vertex of G is a member of D or is adjacent to a member of D. The domination number of G, denoted by cðGÞ, is the cardinality of a smallest dominating set of G. In this paper two cost functions, dqðGÞ and DqðGÞ, are considered which measure respectively the smallest possible and the largest possible increase in the cardinal-ity of a dominating set, over and above cðGÞ,ifq edges were to be removed from G. Bounds are established on dqðGÞ and DqðGÞ for a general graph G, after which these bounds are sharpened or these parameters are determined exactly for a number of special graph classes, including paths, cycles, complete bipartite graphs and complete graphs

Cyber Defense Remediation in Energy Delivery Systems

The integration of Information Technology (IT) and Operational Technology (OT) in Cyber-Physical Systems (CPS) has resulted in increased efficiency and facilitated real-time information acquisition, processing, and decision making. However, the increase in automation technology and the use of the internet for connecting, remote controlling, and supervising systems and facilities has also increased the likelihood of cybersecurity threats that can impact safety of humans and property. There is a need to assess cybersecurity risks in the power grid, nuclear plants, chemical factories, etc. to gain insight into the likelihood of safety hazards. Quantitative cybersecurity risk assessment will lead to informed cyber defense remediation and will ensure the presence of a mitigation plan to prevent safety hazards. In this dissertation, using Energy Delivery Systems (EDS) as a use case to contextualize a CPS, we address key research challenges in managing cyber risk for cyber defense remediation. First, we developed a platform for modeling and analyzing the effect of cyber threats and random system faults on EDS\u27s safety that could lead to catastrophic damages. We developed a data-driven attack graph and fault graph-based model to characterize the exploitability and impact of threats in EDS. We created an operational impact assessment to quantify the damages. Finally, we developed a strategic response decision capability that presents optimal mitigation actions and policies that balance the tradeoff between operational resilience (tactical risk) and strategic risk. Next, we addressed the challenge of management of tactical risk based on a prioritized cyber defense remediation plan. A prioritized cyber defense remediation plan is critical for effective risk management in EDS. Due to EDS\u27s complexity in terms of the heterogeneous nature of blending IT and OT and Industrial Control System (ICS), scale, and critical processes tasks, prioritized remediation should be applied gradually to protect critical assets. We proposed a methodology for prioritizing cyber risk remediation plans by detecting and evaluating critical EDS nodes\u27 paths. We conducted evaluation of critical nodes characteristics based on nodes\u27 architectural positions, measure of centrality based on nodes\u27 connectivity and frequency of network traffic, as well as the controlled amount of electrical power. The model also examines the relationship between cost models of budget allocation for removing vulnerabilities on critical nodes and their impact on gradual readiness. The proposed cost models were empirically validated in an existing network ICS test-bed computing nodes criticality. Two cost models were examined, and although varied, we concluded the lack of correlation between types of cost models to most damageable attack path and critical nodes readiness. Finally, we proposed a time-varying dynamical model for the cyber defense remediation in EDS. We utilize the stochastic evolutionary game model to simulate the dynamic adversary of cyber-attack-defense. We leveraged the Logit Quantal Response Dynamics (LQRD) model to quantify real-world players\u27 cognitive differences. We proposed the optimal decision making approach by calculating the stable evolutionary equilibrium and balancing defense costs and benefits. Case studies on EDS indicate that the proposed method can help the defender predict possible attack action, select the related optimal defense strategy over time, and gain the maximum defense payoffs. We also leveraged software-defined networking (SDN) in EDS for dynamical cyber defense remediation. We presented an approach to aid the selection security controls dynamically in an SDN-enabled EDS and achieve tradeoffs between providing security and Quality of Service (QoS). We modeled the security costs based on end-to-end packet delay and throughput. We proposed a non-dominated sorting based multi-objective optimization framework which can be implemented within an SDN controller to address the joint problem of optimizing between security and QoS parameters by alleviating time complexity at O(MN2). The M is the number of objective functions, and N is the population for each generation, respectively. We presented simulation results that illustrate how data availability and data integrity can be achieved while maintaining QoS constraints

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present