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

Guarding Networks Through Heterogeneous Mobile Guards

In this article, the issue of guarding multi-agent systems against a sequence of intruder attacks through mobile heterogeneous guards (guards with different ranges) is discussed. The article makes use of graph theoretic abstractions of such systems in which agents are the nodes of a graph and edges represent interconnections between agents. Guards represent specialized mobile agents on specific nodes with capabilities to successfully detect and respond to an attack within their guarding range. Using this abstraction, the article addresses the problem in the context of eternal security problem in graphs. Eternal security refers to securing all the nodes in a graph against an infinite sequence of intruder attacks by a certain minimum number of guards. This paper makes use of heterogeneous guards and addresses all the components of the eternal security problem including the number of guards, their deployment and movement strategies. In the proposed solution, a graph is decomposed into clusters and a guard with appropriate range is then assigned to each cluster. These guards ensure that all nodes within their corresponding cluster are being protected at all times, thereby achieving the eternal security in the graph.Comment: American Control Conference, Chicago, IL, 201

Critical Success Factors for Positive User Experience in Hotel Websites: Applying Herzberg's Two Factor Theory for User Experience Modeling

This research presents the development of a critical success factor matrix for increasing positive user experience of hotel websites based upon user ratings. Firstly, a number of critical success factors for web usability have been identified through the initial literature review. Secondly, hotel websites were surveyed in terms of critical success factors identified through the literature review. Thirdly, Herzberg's motivation theory has been applied to the user rating and the critical success factors were categorized into two areas. Finally, the critical success factor matrix has been developed using the two main sets of data.Comment: Journal articl

Secure domination number of kk-subdivision of graphs

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called a secure dominating set of $G$, if for every $u\in V-D$, there exists a vertex $v\in D$ such that $uv \in E$ and $D-\{v\}\cup\{u\}$ is a dominating set of $G$. The cardinality of a smallest secure dominating set of $G$, denoted by $\gamma_s(G)$, is the secure domination number of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the secure domination number of $k$-subdivision of $G$.Comment: 10 Pages, 8 Figure