Secure Domination Research Articles

A Cross-Entropy Approach to the Domination Problem and Its Variants.

The domination problem and three of its variants (total domination, 2-domination, and secure domination) are considered. These problems have various real-world applications, including error correction codes, ad hoc routing for wireless networks, and social network analysis, among others. However, each of them is NP-hard to solve to provable optimality, making fast heuristics for these problems desirable. There are a wealth of highly developed heuristics and approximation algorithms for the domination problem; however, such heuristics are much less common for variants of the domination problem. We redress this gap in the literature by proposing a novel implementation of the cross-entropy method that can be applied to any sensible variant of domination. We present results from experiments that demonstrate that this approach can produce good results in an efficient manner even for larger graphs and that it works roughly as well for any of the domination variants considered.

A lower bound for secure domination number of an outerplanar graph

A subset S of vertices in a graph G is a secure dominating set of G if S is a dominating set of G and, for each vertex u⁄∈S, there is a vertex v∈S such that uv is an edge and (S∖{v})∪{u} is also a dominating set of G. The secure domination number of G, denoted by γs(G), is the cardinality of a smallest secure dominating sets of G. In this paper, we prove that, for any outerplanar graph with n≥4 vertices, γs(G)≥(n+4)/5 and the bound is tight.

On the Strong Secure Domination Number of a Graph

In this paper, we characterize trees with a strong secure domination number less than or equal to 4 and compute this parameter for certain classes of graphs. Also, we investigate bounds for the strong secure domination number of vertex gluing of two graphs.

Secure Inverse Domination in the Corona and Lexicographic Product of Two Graphs

As secure domination and inverse domination garnered attention from various researchers, the combination of the two also raised a certain amount of curiosity. This paper aimed to investigate the secure inverse domination in graphs which is defined as follows. Let G be a connected simple graph and let D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is an inverse dominating set of G with respect to D. The set S is called a secure inverse dominating set of G if for every u∈V(G)∖S, there exists v∈S such that uv∈E(G) and the set (S∖{v})∪{u} is a dominating set of G. The secure inverse domination number of G, denoted by γ_s^((-1) ) (G), is the minimum cardinality of a secure inverse dominating set of G. A secure inverse dominating set of cardinality γ_s^((-1) ) (G) is called γ_s^((-1) )-set. Particularly, the researchers examined and provided the characterization of secure inverse dominating set in the corona and lexicographic product of two graphs in this study. Moreover, the secure inverse domination number of graphs under the binary operations corona and lexicographic product were determined.

Disjoint Perfect Secure Domination in the Cartesian Product and Lexicographic Product of Graphs

Let G be a graph. A dominating set D⊆V(G) is called a secure dominating set of G if for each vertex u∈V(G)∖D, there exists a vertex v∈D such that uv∈ E(G) and the set (D∖{v})∪{u} is a dominating set of G. If every u∈V(G)∖D is adjacent to exactly one vertex in D, then D is a perfect secure dominating set of G. Let D be a minimum perfect secure dominating set of G. If S⊆V(G)∖D is a perfect secure dominating set of G, then S is called an inverse perfect secure dominating set of G with respect to D. A disjoint perfect secure dominating set of G is the set C=D∪S⊆V(G). Furthermore, the disjoint perfect secure domination number, denoted by γ_ps γ_ps (G), is the minimum cardinality of a disjoint perfect secure dominating set of G. A disjoint perfect secure dominating set of cardinality γ_ps γ_ps (G) is called γ_ps γ_ps-set. In this paper, we extended the study on the concept of disjoint perfect secure domination in graphs. Furthermore, we characterized the disjoint perfect secure domination in the Cartesian product and lexicographic product of two graphs.

Identifying Secure Domination in the Cartesian and Lexicographic Products of Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in the Cartesian product and lexicographic product of two connected graphs.

Identifying Secure Domination in the Join and Corona of Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An Identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in join and corona of two graphs.

Disjoint Perfect Secure Domination in the Join and Corona of Graphs

Disjoint Perfect Secure Domination in the Join and Corona of Graphs

Algorithmic aspects of secure domination in unit disk graphs

Given a graph G with vertex set V, a set S⊆V is a secure dominating set of G if S is a dominating set of G and if for every vertex u∈V∖S, there exists a vertex v∈S adjacent to u such that (S∪{u})∖{v} is a dominating set of G.The minimum secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph. In this paper, we first show that the decision version of the MSDS problem is NP-complete in unit disk graphs, even in grid graphs. Secondly, we give an O(n+m) time t-approximation algorithm for the MSDS problem in several geometric intersection graphs which are K1,t-free for some integer t≥3. Finally, we propose a PTAS for the MSDS problem in unit disk graphs.

Domination-related parameters in middle graphs

The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.

2-Secure Domination in Bipolar Hesitancy Fuzzy Graph

We establish secure domination in bipolar hesitancy fuzzy graph. We extend the concept to secure total domination and 2- secure domination in bipolar hesitancy fuzzy graph. Further some theorems and examples related to secure domination are discussed.

Political Reform and the Dilemma of Jordan’s Democratic Position

This study aims at exploring the determinants of political reform, and the problems of democratic repositioning in Jordan. The study will focus on government's relationship with Parliament, and its stand towards Muslim Brotherhood, with security concerns linked to obstacles that stand in the way of the Jordanian democratic reform, along with Jordan’s rank indicators within the international democratic reports. This study was based on the following hypothesis: There is a positive correlation between the regional and international development trends and the deterioration of the economic conditions in Jordan, on the one hand, and the need for more political reform and democratic positioning, on the other hand. To achieve the study purpose, Historical and Descriptive - analytical methods were used. The study concluded that the political reform process has been stagnant, and it has multiple hitches, which may limit the political authority’s capacity toward sound political position. This issue file is being confronted with a lot of obstacles associated with political, constitutional, social, ……etc, that could be summarized as: 1. The absence of a real political determination to achieve political reform. This is due to the dominance of security concerns over the political mind of the Jordanian State. 2. The inability to unify the Jordanian national identity, because of the strength of subsidiary loyalties, at the expense of the national identity. 3. The Jordanian authority still dominates the parliament and there is no influential parliamentary institution. There is no Separation of powers, as the government has the right to issue the temporary laws. The centrality of the king’s role in the Jordanian political system. The King Appoints the Prime Minister and may dismiss him or accept his resignation. He appoints the Ministers; he also dismisses them or accepts their resignation, upon the recommendation of the Prime Minister, without any influential parliamentary role, despite the first parliament in Jordan was elected in 1929 before the declaration of independence. The study also, concluded that the deface of the political landscape has deepened the weakness of political parties, and the weakness of the opposition and electoral laws. The political reform stopped due to the end of the pushing effects. The Arab Spring have contributed to increase the reform, but the subsequent period has seen a marked decreasing in the political reform process. Keywords: Jordan, Political reform, Political positioning, Islamic movement.

On secure domination number of generalized Mycielskian of some graphs

Generalized Mycielskians are triangle-free networks with a large chromatic number, having a number of desirable characteristics like fast multi-path communication, high fault tolerance, reliable resource sharing, etc. Graph invariants like domination number and secure domination number can be used to protect the network by monitoring each node or moving to a failing node. A dominating set of a graph G is a subset of its vertex set, which can monitor every other vertex of the graph, and γ (G), the domination number of G, is the least cardinality among all dominating sets of G. A secure dominating set S of G is a dominating set with an additional property that, for each vertex u not in S there is a vertex v in S adjacent to u such that, the swapped set (S - {v}) ∪ {u} is dominating. γs (G) is the secure domination number of G, which is the minimum cardinality among all secure dominating sets of G. In this paper, we analyzed γ and γs of the generalized Mycielskian (μm) for path graphs Pn and cycle graphs Cn, varying both n and m, and used these results to obtain bounds for general graphs G.

Israeli and the Palestinian present: A temporary arrangement in the West Bank for ethnonational ongoing conflict

This study argues that a political reality of a temporary arrangement has been solidified in Judea and Samaria since 2005, right after Abu Mazen was elected to be the President. It based on integrative approach suggests here for the first time and falls between the political peace process and the armed struggle approaches as a possible solution to the Israeli-Palestinian conflict. This is a result of social-political change within the Palestinian society: the Palestinian Authority has been concerned about maintaining its civil achievements since 2007, based on Fayyad’s program of building Palestinian statehood from the “bottom up.”Israel has used to deepen its security and settlement hold on the ground.This situation guarantees continuation of Israeli political and security dominance in this region; one de facto political entity is being settled between the Mediterranean Sea and the Jordan River, where two distinctly different communities are living side by side.

Secure domination of some graph operators

Secure domination of some graph operators

Secure Domination Cover Pebbling Number of Join of graphs

Objectives: To find the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ). Methods: We define Secure domination cover pebbling number, fsd p(G), of a graph G as the minimum number of pebbles that must be placed on V(G) such that, after a sequence of pebbling moves, the set of vertices with pebbles forms a secure dominating set for G. Findings: We found the secure domination cover pebbling number for the join of two graphs G(p; q) and Kn. Also, the secure domination cover pebbling number for the join of two graphs G(p; q) and G ′ (p ′ ;q ′ ) is determined when the cardinality of the secure dominating set is 2, 3 and 4. A generalization for the secure domination cover pebbling number of path Pn is also found. Subject Mathematics Classification: 05C38, 05C69 Keywords: Graph pebbling; Secure domination; Cover pebbling number; Secure domination cover pebbling number 2010 Subject Mathematics Classification: 05C38; 05C69

Secure equitability in graphs

In secure domination [A. P. Burger, M. A. Henning and J. H. Van Vuuren, Vertex covers and secure domination in graphs, Quaest Math. 31 (2008) 163–171; E. J. Cockayne, Irredundance, secure domination and maximum degree in trees, Discrete Math. 307(1) (2007) 12–17; E. J. Cockayne, O. Favaron and C. M. Mynhardt, Secure domination, weak Roman domination and forbidden subgraph, Bull. Inst. Combin. Appl. 39 (2003) 87–100; C. M. Mynhardt, H. C. Swart and E. Ungerer, Excellent trees and secure domination, Util. Math. 67 (2005) 255–267], a vertex outside has the chance of coming inside the dominating set by replacing an element of the set without affecting domination. This idea is combined with equitability. Secure equitable dominating set is introduced and studied. Other concepts like independence and rigid security are also studied in this paper.

Algorithmic results in secure total dominating sets on graphs

Given a graph G=(V,E), a (total) dominating set of G is a subset D⊆V such that each vertex in V∖D (respectively, V) is adjacent to at least one vertex in D. A (total) dominating set D of G is a secure (total) dominating set if for each v∈V∖D there is a vertex u∈D adjacent to v such that (D∖{u})∪{v} is also a (total) dominating set of G. The minimum cardinality of a secure (total) dominating set of G is called the secure (total) domination number of G. The secure (total) domination problem is to find a minimum secure (total) dominating set of any graph.In this paper, we study the secure total domination problem. We show that the problem restricted to trees is solvable in linear-time, the decision version of the problem is NP-complete for circle graphs, the complexity of the problem differs from that of the secure domination problem, and the problem is APX-complete for graphs of degree at most 4. Furthermore, we show that the optimization version of the problem on bipartite graphs cannot be approximated in polynomial time within (1−ε)ln⁡|V| for any ε>0, unless NP ⊆ DTIME(|V|O(log⁡log⁡|V|)).

Protection of a network by complete secure domination

A complete secure dominating set of a graph $G$ is a dominating set $D \subseteq V(G)$ with the property that for each $v \in D$, there exists $F=\lbrace v_{j} \vert v_{j} \in N(v) \cap (V(G)-D)\rbrace$, such that for each $v_{j} \in F$, $( D-\lbrace v \rbrace) \cup \lbrace v_{j} \rbrace$ is a dominating set. The minimum cardinality of any complete secure dominating set is called the complete secure domination number of $G$ and is denoted by $\gamma_{csd}(G)$. In this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of $G$ are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of $G$.

The secure domination number of Cartesian products of small graphs with paths and cycles

The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Möbius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itself, and it is expected that these methods may be used to determine further results in the future.