Domination Problem Research Articles

Improved Lower Bounds for Queen’s Domination via an Exactly-Solvable Relaxation

The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a m × n chessboard so that they either attack or occupy every cell? We propose a relaxation of the Queen's Domination problem and solve it exactly for rectangular chessboards. As a consequence, we improve on the best known lower bound for rectangular chessboards for one-eighth of the nontrivial cases. As another consequence, we generalize and provide a new interpretation of the best known lower bounds for Queen's Domination of square n × n chessboards for n ≡ {0, 1, 2} mod 4.Finally, we show some results and make some conjectures towards the goal of simplifying the long complicated proof for the best known lower bound for square boards when n ≡ 3 mod 4 (and n > 11). These simply stated conjectures may also be of independent interest.

Power domination in generalized Mycielskian of spiders

The power domination problem in the graph was introduced to model the monitoring problem in electric networks. It is to find a set of vertices with minimum cardinality, called a power dominating set (PDS), that monitors the whole set $ V $ after applying the rules, domination and propagation. In this paper, we discuss the power domination problem in generalized Mycielskian of spiders. We characterize spiders with $\gamma _P(\mu _m(T))= 1$. If $\gamma _P(\mu _m(T))\neq 1 $ and $ m $ is even then $ \gamma _P(\mu _m(T))=\frac {m}{2} +1$. We also show that if $ m $ is odd and $ \gamma _P(\mu _m(T))\neq 1 $, then $ \lceil \frac {m+1}{2}\rceil \leq \gamma _P(\mu _m(T))\leq \lceil \frac {m}{2}\rceil +1$.

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.

The k-th Roman domination problem is polynomial on interval graphs

The k-th Roman domination problem is polynomial on interval graphs

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

Total (restrained) domination in unit disk graphs

The minimum total domination problem and the minimum total restrained domination problem are classical combinatorial optimization problems. In this paper, we first show that the decision versions of both the minimum total domination problem and the minimum total restrained domination problem are NP-complete for grid graphs (a subclass of unit disk graphs), thereby strengthening the result on the total domination decision problem in unit disk graphs established by Jena et al. (2021). Then, we present a linear-time 6.387-approximation algorithm for the minimum total domination problem in unit disk graphs, improving upon the previous algorithm with a ratio of 8 by Jena et al. (2021). Finally, we propose a linear-time 10.387-approximation algorithm and a PTAS for the minimum total restrained domination problem in unit disk graphs.

Theoretical studies of the [formula omitted]-strong Roman domination problem

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 motivated generalization of this problem, known as the k-strong Roman domination. In this problem variant, defenders within a network are tasked with safeguarding any k vertices simultaneously, under multiple attacks. The aim is to find a feasible mapping that assigns a (integer) weight to each vertex of the input graph with a minimum sum of weights across all vertices. A function is considered feasible if any non-defended vertex, i.e. one labeled by zero, is protected by at least one of its neighboring vertices labeled by at least two. Furthermore, each defender ensures the safety of a non-defended vertex by imparting a value of one to it while retaining a one for themselves. To the best of our knowledge, this paper represents the first theoretical study on this problem. The study presents results for general graphs, establishes connections between the problem at hand and other domination problems, and provides exact values and bounds for specific graph classes, including complete graphs, paths, cycles, complete bipartite graphs, grids, and a few selected classes of convex polytopes. Additionally, an attainable lower bound for general cubic graphs is provided.

On maximal Roman domination in graphs: complexity and algorithms

For a simple undirected connected graph G = (V, E), a maximal Roman dominating function (MRDF) of G is a function f : V (G) → {0, 1, 2} with the following properties: (i) For every vertex v ∈ {v ∈ V|f(v) = 0}, there exists a vertex u ∈ N(v) such that f(u) = 2. (ii) The set {v ∈ V|f(v) = 0} is not a dominating set of G; In other words, there exists a vertex v ∈ {v ∈ V|f(v) ≠ 0} such that N(v) ∩ {u ∈ V|f(u) = 0} ∅. The weight of an MRDF of G is the sum of its function values over all vertices, denoted as f(G) = ∑v∈V (G) f(v), and the maximal Roman domination number of G, denoted by γmR(G), is the minimum weight of an MRDF of G. In this paper, we establish some bounds of the maximal Roman domination number of graphs. Additionally, we develop an integer linear programming formulation to compute the maximal Roman domination number of any graph. Furthermore, we prove that maximal Roman domination problem (MRD) is NP-complete even restricted to star convex bipartite graphs and chordal bipartite graphs. Lastly, we show the maximal Roman domination number of threshold graphs, trees, and block graphs can be computed in linear time.

Introducing 3-Path Domination in Graphs

The dominating set of a graph G is a set of vertices D such that for every v ∈ V ( G ) either v ∈ D or v is adjacent to a vertex in D . The domination number, denoted γ ( G ) , is the minimum number of vertices in a dominating set. In 1998, Haynes and Slater [1] introduced paired-domination. Building on paired-domination, we introduce 3-path domination. We define a 3-path dominating set of G to be D = { Q 1 , Q 2 , … , Q k | Q i is a 3-path } such that the vertex set V ( D ) = V ( Q 1 ) ∪ V ( Q 2 ) ∪ ⋯ ∪ V ( Q k ) is a dominating set. We define the 3-path domination number, denoted by γ P 3 ( G ) , to be the minimum number of 3-paths needed to dominate G . We show that the 3-path domination problem is NP-complete. We also prove bounds on γ P 3 ( G ) and improve those bounds for particular families of graphs such as Harary graphs, Hamiltonian graphs, and subclasses of trees. In general, we prove γ P 3 ( G ) ≤ n 3 .

An algorithm for the secure total domination problem in proper interval graphs

A subset S of vertices of G is a total dominating set if, for any vertex v, there is a vertex in S adjacent to v. A total dominating set S is a secure total dominating set if, for any vertex v∉S, there is a vertex u∈S such that uv is an edge and (S∖{u})∪{v} is also a total dominating set. In this paper, we design an O(m)-time algorithm for computing a minimum secure total dominating set in a proper interval graph, where m is the number of edges.

An extension of locating-total domination problem and its complexity

An extension of locating-total domination problem and its complexity

On double Roman domination problem for several graph classes

On double Roman domination problem for several graph classes

Metaheuristic algorithms for solving roman {2}-domination problem

A Roman {2}-dominating function (Rom2DF) on a graph G(V, E) is a function g : V → {0, 1, 2} of G such that for every vertex x ∈ V with g(x) = 0, either there exists a neighbor y of x with g(y) = 2 or at least two neighbors, u, v with g(u) = g(v) = 1. The value w(g) = ∑x∈V g(x) is the weight of the Rom2DF. The minimum weight of a Rom2DF of G is called the Roman {2}-domination number denoted by γ{R2}(G). Since determining γ{R2}(G) of a graph G is NP-hard and no metaheuristic algorithms have been proposed for the same, two procedures based on genetic algorithm are proposed as a solution for the Roman {2}-domination problem. One of the proposed methods employs a random initial population, while the other uses a population generated using heuristics. Experiments have been carried out on graphs generated using Erdös–Rényi model, a popular model for graph generation and Harwell Boeing (HB) dataset. The experimental results demonstrate that both approaches provide a near optimal solution which is well within the known lower and upper bounds for the problem. The experimental results further show that the procedure based on random initial population has outperformed the heuristic based procedure.

Generalized domination of ergodic elements in ordered Banach algebras

Let A be an ordered Banach algebra and a, b ∈ A such that 0 ≤ a ≤ b. The domination problem involves finding natural conditions under which properties of b will be inherited by a. In [12] this problem was investigated for ergodic elements, where a ∈ A is ergodic if the sequence converges in A. In this paper we consider the more general problem where the assumption 0 ≤ a ≤ b is replaced by the weaker condition ±a ≤ b.

Global Diversity Character Habituation Profil Pelajar Pancasila in Islamic-Based Schools from the Perspective of Pierre Bourdieu and Clifford Geertz

This research is based on the phenomenon of conflict with a background of differences. Indonesia, as a multicultural country that has the ideology of Pancasila and Bhinneka Tunggal Ika, in fact still often experiences discrimination, domination, and other problems that end in conflict and violence. Indonesia's cultural wealth in the form of local wisdom is now starting to be eroded due to the influence of globalization. Society began to focus on Western cultures. This research aims to describe the process of habituating the character of global diversity in the younger generation, especially students through educational institutions. Based on the theory of Pierre Bourdieu and Clifford Geertz, the researcher tries to describe the implementation of strengthening the global diversity character of the Profil Pelajar Pancasila in Islamic-based schools in the city of Surakarta. This research is research with a descriptive qualitative approach. The results of this research explain that Profil Pelajar Pancasila is a competent lifelong learner, has character, and behaves according to Pancasila's values. In connection with Pierre Bourdieu's theory, educational institutions are referred to as arenas (fields). Furthermore, global diversity includes openness and tolerance towards foreign cultural diversity while still respecting culture. This concept is by Clifford Geertz's theory that culture is a symbolic system, so cultural processes must be read, translated, and interpreted. The success of characterizing global diversity is influenced by family socialization and teacher example, both of which, according to Pierre Bourdieu, are structural actors/agents and are influenced by social heterogeneity which becomes social capital.

Algorithmic aspect on total Roman {2}-domination of Cartesian products of paths and cycles

A total Roman {2}-dominating function (TR2DF) on a graph G with vertex set V is a function f : V → {0, 1, 2} having the property that for every vertex v with f(v) = 0, ∑u∈N (v) f(u) ≥ 2, where N(v) represents the open neighborhood of v, and the subgraph of G induced by the set of vertices with f(v) > 0 has no isolated vertex. The weight of a TR2DF f is the value w(f) = ∑v∈V f(v), and the minimum weight of a TR2DF of G is the total Roman {2}-domination number γtR2(G). The total Roman {2}-domination problem (TR2DP) is to determine the value γtR2(G). In this paper, we first propose an integer linear programming (ILP) formulation for the TR2DP. Furthermore, we apply the discharging approach to determine the total Roman {2}-domination number for some Cartesian products of paths and cycles.

A knowledge-based iterated local search for the weighted total domination problem

For a simple undirected weighted graph G=(V,E,w,c), the weighted total domination problem is to find a total dominating set S with the minimum weight cost. A total dominating set S is a vertex subset satisfying that for each vertex in V there is at least one neighboring vertex in S. We propose a knowledge-based iterated local search algorithm for this problem that combines a reduction procedure to reduce the input graph, a learning-based initialization to generate high-quality initial solutions and a solution-based iterated local search to conduct intensive solution examination. Experiments on 342 benchmark instances show that the algorithm outperforms state-of-the-art algorithms. In particular, it reports 93 new upper bounds and 249 same results (including 165 known optimal results). The impact of each component of the algorithm is examined.

Minimal Roman Dominating Functions: Extensions and Enumeration

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman dominating functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document} and a function f:V→{0,1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:V\\rightarrow \\{0,1,2\\}$$\\end{document}, is there a minimal Roman dominating function f~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ilde{f}$$\\end{document} with f≤f~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\le \ ilde{f}$$\\end{document}? Here, ≤\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\le $$\\end{document} lifts 0<1<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0< 1< 2$$\\end{document} pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of O(1.9332n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(1.9332^n)$$\\end{document} for graphs of order n; this is complemented by a lower bound example of Ω(1.7441n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (1.7441^n)$$\\end{document}.

Some new algorithmic results on co-secure domination in graphs

For a simple graph G=(V,E) without any isolated vertex, a co-secure dominating set S of G satisfies two properties, (i) S is a dominating set of G, (ii) for every vertex v∈S, there exists a vertex u∈V∖S such that uv∈E and (S∖{v})∪{u} is a dominating set of G. The minimum cardinality of a co-secure dominating set of G is called the co-secure domination number of G and is denoted by γcs(G). The Co-secure Domination problem is to find a co-secure dominating set of a graph G of cardinality γcs(G). The decision version of the problem is known to be NP-complete for bipartite, planar, and split graphs. On the other hand, it is also known that the Co-secure Domination problem is efficiently solvable for proper interval graphs and cographs.In an effort to reduce the complexity gap of the Co-secure Domination problem, in this paper, we work on various important graph classes. We show that the decision version of the problem remains NP-complete for doubly chordal graphs and for some subclasses of bipartite graphs, namely, chordal bipartite graphs, star-convex bipartite graphs, and comb-convex bipartite graphs. On the positive side, we give an efficient algorithm to compute the co-secure domination number of chain graphs, which is an important subclass of bipartite graphs. In addition, we show that the problem is linear-time solvable for bounded tree-width graphs and bounded clique-width graphs. Further, we establish that the computational complexity of this problem differs from that of the classical domination problem.

A Survey on Variant Domination Problems in Geometric Intersection Graphs

Given a graph [Formula: see text] with vertex set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if every vertex is either in [Formula: see text] or is adjacent to a vertex in [Formula: see text]. There are a lot of variants for dominating sets, such as connected dominating sets, total dominating sets, total restricted dominating sets, secure (connected, total) dominating sets, etc. Geometric intersection graphs are graphs for which there is a bijection [Formula: see text] between the vertices and a set [Formula: see text] of geometric objects (for example, disks, rectangles, etc.) such that there is an edge between two vertices [Formula: see text] and [Formula: see text] if and only if the objects [Formula: see text] and [Formula: see text] intersect. In this paper, we offer a survey about complexity and algorithmic results on variant domination problems in geometric intersection graphs.