Domination Problem Research Articles

On Signed Star Domination in Graphs

For a graph G = (V, E) without isolated vertex, a function f: E(G) → {−1, 1} is said to be a signed star dominating function of G\(\sum\limits_{e \in E(v)} {f(e) \ge 1} \) for every v ∈ V(G), where E(v) = {uv} ∈ E(G)∣ u ∈ V(G)}. The minimum value of \(\sum\limits_{e \in E(G)} {f(e)} \), taken over all signed star dominating functions f of G, is called the signed star domination number of G and is denoted by γss(G). This paper studies the bounds and algorithms of signed star domination numbers in some classes of graphs. In particular, sharp bounds for the signed star domination number of a general graph and a linear-time algorithm for the signed star domination problem in a tree is presented.

Roman domination problem with uncertain positioning and deployment costs

In a connected simple graph, the weighted Roman domination problem is considered at which the cost of positioning at each vertex is imposed in addition to the costs of potential deployments from a vertex to some of its neighboring vertices. Proper decision in practice is prone to a high degree of indeterminacy, mostly raised by unpredictable events that do not obey the rules and prerequisites of the probability theory. In this study, we model this problem with such assumptions in the context of the uncertainty theory initiated by Liu (Uncertainty theory. Studies in fuzziness and soft computing, Springer, Berlin, 2007). Two different optimization models are presented, and a concrete example is provided for illustrative purposes. Weaknesses of the probability theory and fuzzy theory in dealing with this problem are also mentioned in detail.

On directed covering and domination problems

In this paper, we study covering and domination problems on directed graphs. Although undirected Vertex Cover and Edge Dominating Set are well-studied classical graph problems, the directed versions have not been studied much due to the lack of clear definitions.We give natural definitions for Directedr-In (Out) Vertex Cover and Directed(p,q)-Edge Dominating Set as directed generalizations of Vertex Cover and Edge Dominating Set. For these problems, we show that (1) Directedr-In (Out) Vertex Cover and Directed(p,q)-Edge Dominating Set are NP-complete on planar directed acyclic graphs except when r=1 or (p,q)=(0,0), (2) if r≥2, Directedr-In (Out) Vertex Cover is W[2]-hard and clnk-inapproximable on directed acyclic graphs, (3) if either p or q is greater than 1, Directed(p,q)-Edge Dominating Set is W[2]-hard and clnk-inapproximable on directed acyclic graphs, (4) all problems can be solved in polynomial time on trees, and (5) Directed(0,1)-Edge ((1,0)-Edge, (1,1)-Edge) Dominating Set is fixed-parameter tractable on general graphs.The first result implies that Directedr-Dominating Set on directed line graphs is NP-complete even if r=1.

The secure domination problem in cographs

Let G=(V,E) be a graph. A subset D of vertices of G is called a dominating set of G if for every u∈V∖D, there exists a vertex v∈D such that uv∈E. A dominating set D of a graph G is called a secure dominating set of G if for every u∈V∖D, there exists a vertex v∈D such that uv∈E and (D∖{v})∪{u} is a dominating set of G. The secure domination number of G, denoted by γs(G), is the minimum cardinality of a secure dominating set of G. In this paper, we present an O(n+m) time algorithm to compute the secure domination number of a cograph having n vertices and m edges.

Variable Neighborhood Search Approach for Solving Roman and Weak Roman Domination Problems on Graphs

In this paper Roman and weak Roman domination problems on graphs are considered. Given that both problems are NP hard, a new heuristic approach, based on a Variable Neighborhood Search (VNS), is presented. The presented algorithm is tested on instances known from the literature, with up to 600 vertices. The VNS approach is justified since it was able to achieve an optimal solution value on the majority of instances where the optimal solution value is known. Also, for the majority of instances where optimization solvers found a solution value but were unable to prove it to be optimal, the VNS algorithm achieves an even better solution value.

The semitotal domination problem in block graphs

The semitotal domination problem in block graphs

Recent Results on the Power Domination Numbers of Graph Products

We review recent results on the power domination problem of graph products and establish improved results for some families of graph products, namely, , , Pn ⊠ Pm, Pn ⊠ Cm and Cn ⊠ Cm. We also characterize graphs G and H for which the power domination number of the Cartesian product of G and H, which is denoted as , is 1.

Zero forcing number, Grundy domination number, and their variants

This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem, providing a linear algebra approach to the domination problem. We show that several Grundy domination type parameters are bounded above by minimum rank type parameters. We also give a method to calculate the L-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the L-Grundy domination number.

Signed double Roman domination in graphs

A signed double Roman dominating function (SDRDF) on a graph G=(V,E) is a function f:V(G)→{−1,1,2,3} such that (i) every vertex v with f(v)=−1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w)=3, (ii) every vertex v with f(v)=1 is adjacent to at least one vertex w with f(w)≥2 and (iii) ∑u∈N[v]f(u)≥1 holds for any vertex v. The weight of an SDRDF f is ∑u∈V(G)f(u), the minimum weight of an SDRDF is the signed double Roman domination number γsdR(G) of G. In this paper, we prove that the signed double Roman domination problem is NP-complete for bipartite and chordal graphs. We also prove that for any tree T of order n≥2, −5n+249≤γsdR(T)≤n and we characterize all trees attaining each bound.

Dominating induced matchings in [formula omitted]-free graphs

Let G=(V,E) be a finite undirected graph without loops and multiple edges. A subset M⊆E of edges is a dominating induced matching (d.i.m. ) in G if every edge in E is intersected by exactly one edge of M. In particular, this means that M is an induced matching, and every edge not in M shares exactly one vertex with an edge in M. Clearly, not every graph has a d.i.m.The Dominating Induced Matching (DIM) problem asks for the existence of a d.i.m. in G; this problem is also known as the Efficient Edge Domination problem; it is the Efficient Domination problem for line graphs.The DIM problem is NP-complete in general, and even for very restricted graph classes such as planar bipartite graphs with maximum degree 3. However, DIM is solvable in polynomial time for claw-free (i.e., S1,1,1-free) graphs, for S1,2,3-free graphs, for S2,2,2-free graphs as well as for S2,2,3-free graphs, in linear time for P7-free graphs, and in polynomial time for P8-free graphs (Pk is a special case of Si,j,ℓ). In a paper by Hertz, Lozin, Ries, Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial time for Si,j,k-free graphs for every fixed i,j,k.In this paper, combining two distinct approaches, we solve it in polynomial time for S1,2,4-free graphs which generalizes the S1,2,3-free as well as the P7-free case.

FPT algorithms for domination in sparse graphs and beyond

We study the parameterized complexity of domination problems on sparse graphs and beyond. The nowhere dense classes of graphs have been proposed as the main model for sparseness that can be utilized algorithmically. The class of d-degenerate graphs is not nowhere dense, yet domination remains fixed-parameter tractable. Both nowhere dense classes of graphs and d-degenerate graph classes are biclique-free classes, meaning there is an integer t such that no graph in the class contains Kt,t as a subgraph. In this paper we show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs. Our algorithms are simple and rely on results from extremal graph theory that bound the number of edges in a t-biclique free graph. In particular, the problems k-Dominating Set, Connectedk-Dominating Set, Independentk-Dominating Set and Minimum Weightk-Dominating Set are shown to be FPT, when parameterized by t+k, on graphs not containing Kt,t as a subgraph. With the exception of Connectedk-Dominating Set all described algorithms are linear in the size of the input graph.

A dichotomy for weighted efficient dominating sets with bounded degree vertices

In a finite undirected graph G, a vertex v dominates itself and its neighbors. A vertex set D is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is well known to be NP-complete for graphs of maximum degree 3. We say that an e.d.s. D of G is k-bounded (k-b.e.d.s. for short) if the degree of every vertex in D is at most k in G. The task of the k-Bounded Weighted Efficient Domination (k-BWED) problem is to determine whether a given vertex-weighted graph G admits a k-b.e.d.s., and if so, to compute one of minimum weight. It easily follows from the NP-completeness of ED for graphs of maximum degree 3 that the k-BWED problem is NP-complete for every k≥3, and clearly, the k-BWED problem is solvable in linear time for k≤1. In this note, we show that the 2-BWED problem is solvable in time O(|V(G)|(|V(G)|+|E(G)|)), thus obtaining a dichotomy of the complexity status of k-BWED over all k≥0.

Modelling and solving the perfect edge domination problem

A formulation is proposed for the perfect edge domination problem and some exact algorithms based on it are designed and tested. So far, perfect edge domination has been investigated mostly in computational complexity terms. Indeed, we could find no previous explicit mathematical formulation or exact algorithm for the problem. Furthermore, testing our algorithms also represented a challenge. Standard randomly generated graphs tend to contain a single perfect edge dominating solution, i.e., the trivial one, containing all edges in the graph. Accordingly, some quite elaborated procedures had to be devised to have access to more challenging instances. A total of 736 graphs were thus generated, all of them containing feasible solutions other than the trivial ones. Every graph giving rise to a weighted and a non weighted instance, all instances solved to proven optimality by two of the algorithms tested.

Algorithmic aspects of semitotal domination in graphs

For a graph G=(V,E), a set D⊆V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semitotal dominating set of cardinality at most k. The Semitotal Domination Decision problem is known to be NP-complete for general graphs. In this paper, we show that the Semitotal Domination Decision problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs. We give a polynomial time algorithm to solve the Minimum Semitotal Domination problem in interval graphs. We show that the Minimum Semitotal Domination problem in a graph with maximum degree Δ admits an approximation algorithm that achieves the approximation ratio of 2+3ln⁡(Δ+1), showing that the problem is in the class log-APX. We also show that the Minimum Semitotal Domination problem cannot be approximated within (1−ϵ)ln⁡|V| for any ϵ>0 unless NP ⊆ DTIME (|V|O(log⁡log⁡|V|)). Finally, we prove that the Minimum Semitotal Domination problem is APX-complete for bipartite graphs with maximum degree 4.

Polynomial-time algorithm for weighted efficient domination problem on diameter three planar graphs

A set D of vertices in a graph G is an efficient dominating set (e.d.s. for short) of G if D is an independent set and every vertex not in D is adjacent to exactly one vertex in D. The efficient domination (ED) problem asks for the existence of an e.d.s. in G. The minimum weighted efficient domination problem (MIN-WED for short) is the problem of finding an e.d.s. of minimum weight in a given vertex-weighted graph. Brandstädt, Fičur, Leitert and Milanič (2015) [3] stated the running times of the fastest known polynomial-time algorithms for the MIN-WED problem on some graphs classes by using a Hasse diagram.In this paper, we update this Hasse diagram by showing that, while for every integer d such that d=3k or d=3k+2, where k≥1, the ED problem remains NP-complete for graphs of diameter d, the weighted version of the problem is solvable in time O(|V(G)|+|E(G)|) in the class of diameter three bipartite graphs and in time O(|V(G)|5) in the class of diameter three planar graphs.

PROBLEMATIKA PEMBELAJARAN SEJARAH KEBUDAYAAN ISLAM DI MADRASAH TSANAWIYAH AL-KHAIRAAT PAKULI KABUPATEN SIGI

The aim of learning SKI in Madrasah Tsanawiyah is the provision of knowledge about Islamic history and culture to students and taking lessons, values and meanings contained in history. In fact, after being traced, learning Islamic Cultural History in the Al-Khairaat Pakuli Madrasahs of Sigi District faces several problems. The research method used is a qualitative approach, then to obtain data in accordance with the problems, data collection techniques are carried out by observation, interviews, and documentation. The results of the study show that the forms of problematic learning about the history of Islamic culture in Madrasah Tsanawiyah Al-Khairaat Pakuli are; 1) Student Participation Problems, 2) Problems in SKI Teaching Staff, 3) Facilities and Infrastructure Problems, 4) Time or Lesson Hours Problems are very short, 5) Negative Views on SKI Subjects, 6) Domination Problems in Cognitive Aspects of Learning SKI, 7) Problems in Learning Methods that are monotonous.

Complexity of k-tuple total and total {k}-dominations for some subclasses of bipartite graphs

We consider two variations of graph total domination, namely, k-tuple total domination and total {k}-domination (for a fixed positive integer k). Their related decision problems are both NP-complete even for bipartite graphs. In this work, we study some subclasses of bipartite graphs. We prove the NP-completeness of both problems (for every fixed k) for bipartite planar graphs and we provide an APX-hardness result for the total domination problem for bipartite subcubic graphs. In addition, we introduce a more general variation of total domination (total (r,m)-domination) that allows us to design a specific linear time algorithm for bipartite distance-hereditary graphs. In particular, it returns a minimum weight total {k}-dominating function for bipartite distance-hereditary graphs.

Weighted efficient domination for some classes of [formula omitted]-free and of [formula omitted]-free graphs

A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for claw-free graphs, for 2P3-free chordal graphs (and thus, for P7-free graphs), and for bipartite graphs. For the complexity of ED and its weighted version WED, a dichotomy for H-free graphs was reached: A graph H is called a linear forest if H is acyclic and claw-free, that is, if all its components are paths. Thus, the ED problem remains NP-complete for H-free graphs whenever H is not a linear forest. For every linear forest H, WED is either solvable in polynomial time or NP-complete for H-free graphs; the final result showed that WED is solvable in polynomial time for P6-free graphs.For (H1,H2)-free graphs, however, we are still far away from a dichotomy result. The main topics of this paper are: (1) to improve the time bounds and simplify the proofs (based on modular decomposition) for polynomial time cases of WED for some H-free graph classes; (2) to investigate the complexity of WED for some cases of (H1,H2)-free graphs such that WED is NP-complete for Hi-free graphs for at least one of i∈{1,2}. Since it is well known that WED is solvable in polynomial time for graph classes of bounded clique-width, we consider only classes of (H1,H2)-free graphs with unbounded clique-width.

On interval number in cycle convexity

Recently, Araujo et al. [Manuscript in preparation, 2017] introduced the notion of Cycle Convexity of graphs. In their seminal work, they studied the graph convexity parameter called hull number for this new graph convexity they proposed, and they presented some of its applications in Knot theory. Roughly, the tunnel number of a knot embedded in a plane is upper bounded by the hull number of a corresponding planar 4-regular graph in cycle convexity. In this paper, we go further in the study of this new graph convexity and we study the interval number of a graph in cycle convexity. This parameter is, alongside the hull number, one of the most studied parameters in the literature about graph convexities. Precisely, given a graph G, its interval number in cycle convexity, denoted by $in_{cc} (G)$, is the minimum cardinality of a set S ⊆ V (G) such that every vertex w ∈ V (G) S has two distinct neighbors u, v ∈ S such that u and v lie in same connected component of G[S], i.e. the subgraph of G induced by the vertices in S. In this work, first we provide bounds on $in_{cc} (G)$ and its relations to other graph convexity parameters, and explore its behavior on grids. Then, we present some hardness results by showing that deciding whether $in_{cc} (G)$ ≤ k is NP-complete, even if G is a split graph or a bounded-degree planar graph, and that the problem is W[2]-hard in bipartite graphs when k is the parameter. As a consequence, we obtain that $in_{cc} (G)$ cannot be approximated up to a constant factor in the classes of split graphs and bipartite graphs (unless P = N P ). On the positive side, we present polynomial-time algorithms to compute $in_{cc} (G)$ for outerplanar graphs, cobipartite graphs and interval graphs. We also present fixed-parameter tractable (FPT) algorithms to compute it for (q, q − 4)-graphs when q is the parameter and for general graphs G when parameterized either by the treewidth or the neighborhood diversity of G. Some of our hardness results and positive results are not known to hold for related graph convexities and domination problems. We hope that the design of our new reductions and polynomial-time algorithms can be helpful in order to advance in the study of related graph problems.

Trace Ratio Optimization With Feature Correlation Mining for Multiclass Discriminant Analysis

Fisher's linear discriminant analysis is a widely accepted dimensionality reduction method, which aims to find a transformation matrix to convert feature space to a smaller space by maximising the between-class scatter matrix while minimising the within-class scatter matrix. Although the fast and easy process of finding the transformation matrix has made this method attractive, overemphasizing the large class distances makes the criterion of this method suboptimal. In this case, the close class pairs tend to overlap in the subspace. Despite different weighting methods having been developed to overcome this problem, there is still a room to improve this issue. In this work, we study a weighted trace ratio by maximising the harmonic mean of the multiple objective reciprocals. To further improve the performance, we enforce the l2,1-norm to the developed objective function. Additionally, we propose an iterative algorithm to optimise this objective function. The proposed method avoids the domination problem of the largest objective, and guarantees that no objectives will be too small. This method can be more beneficial if the number of classes is large. The extensive experiments on different datasets show the effectiveness of our proposed method when compared with four state-of-the-art methods.