Domination Problem Research Articles

Independent Rainbow Domination of Graphs

Given a positive integer t and a graph F, the goal is to assign a subset of the color set \(\{1,2,\ldots ,t\}\) to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function (\(t\mathrm{RDF}\)) of the graph F. A \(t\mathrm{RDF}\) is independent (\(It\mathrm{RDF}\)) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a \(t\mathrm{RDF}\)g of a graph F is the value \(w(g) =\sum _{v \in V(F)}|g(v)|\). The independent t-rainbow domination number \(i_{rt}(F)\) is the minimum weight over all \(It\mathrm{RDF}\)s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

The 2-Rainbow Domination of Sierpiński Graphs and Extended Sierpiński Graphs

Let G(V, E) be a connected and undirected graph with n-vertex-set V and m-edge-set E. For each v ∈ V, let N(v) = {u|v ∈ V and(u, v) ∈ E}. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to a k-bit Boolean string f(v) = f k (v)f k − 1(v) … f 1(v), i.e., f i (v) ∈ {0, 1}, 1 ≤ i ≤ k, such that for any vertex v with f(v) = 0(k) we have ⋈ u ∈ N(v) f(u) = 1(k), for all v ∈ V, where ⋈ u ∈ S f(u) denotes the result of taking bitwise OR operation on f(u), for all u ∈ S. The weight of f is defined as $w(f) = {\sum }_{v\in V}{\sum }^{k}_{i=1} f_{i}(v)$ . The k-rainbow domination number γ k r (G) is the minimum weight of a k-rainbow dominating function over all k-rainbow dominating functions of G. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G. In this paper, we determine γ 2r (S(n, m)), γ 2r (S +(n, m)), and γ 2r (S ++(n, m)), where S(n, m), S +(n, m), and S ++(n, m) are Sierpinski graphs and extended Sierpinski graphs.

Efficient domination for classes of [formula omitted]-free graphs

In a finite undirected graph G, a vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. 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. in G, is known to be NP-complete even for very restricted graph classes such as P7-free chordal graphs; its complexity was an open question for P6-free graphs and was open even for the subclass of P6-free chordal graphs. Recently, Lokshtanov et al., and independently, Brandstädt and Mosca showed that ED is solvable in polynomial time for P6-free graphs.The ED problem on a graph G can be reduced to the Maximum Weight Independent Set problem on the square of G. In this paper, we show that squares of P6-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P6, house, hole, domino)-free graphs. Thus, ED/WeightedED is solvable in polynomial time for (P6, house, hole, domino)-free graphs, and in particular, for P6-free chordal graphs. Also, the time bound achieved for ED on this class is much better than in the P6-free case.Moreover, we show that squares of P6-free graphs that have an e.d. are hole-free. Based on this result, we show that ED is solvable in polynomial time for (P6, net)-free graphs; again, the time bound for ED on this class is much better than in the P6-free case.

Algorithmic aspects of b-disjunctive domination in graphs

For a graph \(G=(V,E)\), a set \(D\subseteq V\) is called a disjunctive dominating set of G if for every vertex \(v\in V\setminus D\), v is either adjacent to a vertex of D or has at least two vertices in D at distance 2 from it. The cardinality of a minimum disjunctive dominating set of G is called the disjunctive domination number of graph G, and is denoted by \(\gamma _{2}^{d}(G)\). The Minimum Disjunctive Domination Problem (MDDP) is to find a disjunctive dominating set of cardinality \(\gamma _{2}^{d}(G)\). Given a positive integer k and a graph G, the Disjunctive Domination Decision Problem (DDDP) is to decide whether G has a disjunctive dominating set of cardinality at most k. In this article, we first propose a polynomial time algorithm for MDDP in proper interval graphs. Next we tighten the NP-completeness of DDDP by showing that it remains NP-complete even in chordal graphs. We also propose a \((\ln (\Delta ^{2}+\Delta +2)+1)\)-approximation algorithm for MDDP, where \(\Delta \) is the maximum degree of input graph \(G=(V,E)\) and prove that MDDP can not be approximated within \((1-\epsilon ) \ln (|V|)\) for any \(\epsilon >0\) unless NP \(\subseteq \) DTIME\((|V|^{O(\log \log |V|)})\). Finally, we show that MDDP is APX-complete for bipartite graphs with maximum degree 3.

More results on the complexity of domination problems in graphs

Given a graph G = (V, E) and an integer r ≥ 1, we call 'r-dominating code' any subset C of V such that every vertex in V is at distance at most r from at least one vertex in C. We investigate and locate in the complexity classes of the polynomial hierarchy, several problems linked with domination in graphs, such as, given r and G, the existence of, or search for, optimal r-dominating codes in G, or optimal r-dominating codes in G containing a subset of vertices X ⊂ V .

More results on the complexity of domination problems in graphs

Given a graph G = (V, E) and an integer r ≥ 1, we call 'r-dominating code' any subset C of V such that every vertex in V is at distance at most r from at least one vertex in C. We investigate and locate in the complexity classes of the polynomial hierarchy, several problems linked with domination in graphs, such as, given r and G, the existence of, or search for, optimal r-dominating codes in G, or optimal r-dominating codes in G containing a subset of vertices X ⊂ V .

An Eternal Domination Problem in Grids

A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard. Other guards are allowed to move at the same time, and before and after each attack, the vertices containing guards must form a dominating set of the graph. The minimum number of guards that can defend the graph against such an arbitrary sequence of attacks is called the m-eviction number of the graph. In this paper, the m-eviction number is determined exactly for $m \times n$ grids with $m \leq 4$ and upper bounds are given for all $n \geq m \geq 8$.

Formula omitted] massive axions, domain walls and inflation

We have analyzed a U(1) model which is broken explicitly to a Z2 model. The proposal results in generating two types of stable domain walls, in contrast with the more common NDW=1 version which is already used to explain axion invisibility for the UPQ(1) model. We have tried to take into account any possible relation with previous studies. We have studied some of the domain properties, proposing an approximate solution which satisfies boundary conditions and the static virial theorem, simultaneously. Invoking the mentioned approximation, we have been able to obtain an analytical insight about the effect of parameters on the domain wall features, particularly on their surface energy density which is of great importance in cosmological studies when one tries to avoid domain wall energy domination problem. Next, we have mainly focused on the likely inflationary scenarios resulting from the model, including saddle point inflation, again insisting on analytical discussions to be able to follow the role of parameters. We have tried to relate inflationary scenarios to the known categories to take advantage of the previous detailed studies under the inflationary topic over the decades. We have concluded that any successful inflationary scenario requires large fields within the present model. Calculations are mainly done analytically, although numerical results are also obtained to reinforce the analytical results.

Italian domination in trees

The Roman domination number and Italian domination number (also known as the Roman {2}-domination number) are graph labeling problems in which each vertex is labeled with either 0, 1, or 2. In the Roman domination problem, each vertex labeled 0 must be adjacent to at least one vertex labeled 2. In the Italian domination problem, each vertex labeled 0 must have the labels of the vertices in its closed neighborhood sum to at least two. The Italian domination number, γI(G), of a graph G is the minimum possible sum of such a labeling, where the sum is taken over all the vertices in G. It is known that if T is a tree with at least two vertices, then γ(T)+1≤γI(T)≤2γ(T). In this paper, we characterize the trees T for which γ(T)+1=γI(T), and we characterize the trees T for which γI(T)=2γ(T).

Complexity of the game domination problem

The game domination number is a graph invariant that arises from a game, which is related to graph domination in a similar way as the game chromatic number is related to graph coloring. In this paper we show that deciding whether the game domination number of a graph is bounded by a given integer is PSPACE-complete. This contrasts the situation of the game coloring problem whose complexity is still unknown.

Linear kernels for [formula omitted]-tuple and liar’s domination in bounded genus graphs

A set D⊆V is called a k-tuple dominating set of a graph G=(V,E) if |NG[v]∩D|≥k for all v∈V, where NG[v] denotes the closed neighborhood of v. A set D⊆V is called a liar’s dominating set of a graph G=(V,E) if (i) |NG[v]∩D|≥2 for all v∈V and (ii) for every pair of distinct vertices u,v∈V, |(NG[u]∪NG[v])∩D|≥3. Given a graph G, the decision versions of k-Tuple Domination Problem and the Liar’s Domination Problem are to check whether there exist a k-tuple dominating set and a liar’s dominating set of G of a given cardinality, respectively. These two problems are known to be NP-complete (Liao and Chang, 2003; Slater, 2009). In this paper, we study the parameterized complexity of these problems. We show that the k-Tuple Domination Problem and the Liar’s Domination Problem are W[2]-hard for general graphs. It can be verified that both the problems have a finite integer index and satisfy certain coverability property. Hence they admit linear kernel as per the meta-theorem in Bodlaender (2009), but the meta-theorem says nothing about the constant. In this paper, we present a direct proof of the existence of linear kernel with small constants for both the problems.

Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi: 10.1007/s11117-016-0401-9 , 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.

Domination problem in Banach lattices

The objective of this paper is to present a survey of the main results concerning the domination problem for operators in Banach lattices, to lay down a general approach to the study of the problem, and to indicate several directions for further investigations.

Finding Dominating Induced Matchings in $$P_8$$ P 8 -Free Graphs in Polynomial Time

Let $$G=(V,E)$$G=(V,E) be a finite undirected graph. An edge set $$E' \subseteq E$$E?⊆E is a dominating induced matching (d.i.m.) in G if every edge in E is intersected by exactly one edge of $$E'$$E?. 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. The DIM problem is related to parallel resource allocation problems, encoding theory and network routing. It is $${\mathbb {NP}}$$NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree three and is solvable in linear time for $$P_7$$P7-free graphs. However, its complexity was open for $$P_k$$Pk-free graphs for any $$k \ge 8$$k?8; $$P_k$$Pk denotes the chordless path with k vertices and $$k-1$$k-1 edges. We show in this paper that the weighted DIM problem is solvable in polynomial time for $$P_8$$P8-free graphs.

On the complexity of the labeled domination problem in graphs

AbstractIn 2008, a unified approach (labeled domination) to several domination problems (k‐tuple domination, ‐domination, and M‐domination, among others) was introduced. The labeled domination problem is to find an L‐dominating function of minimum weight in a graph. It is an NP‐complete problem even when restricted to split graphs and bipartite graphs. On the other hand, it is known to be polynomial‐time solvable for the class of strongly chordal graphs. In this paper, we state explicit formulas that relate the domination numbers considered. These relationships allow us to enlarge the family of graphs where the labeled domination problem is polynomial‐time solvable to the class of graphs having cliquewidth bounded by a constant.

(Total) Vector domination for graphs with bounded branchwidth

Given a graph G=(V,E) of order n and an n-dimensional non-negative vector d=(d(1),d(2),…,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S⊆V such that every vertex v in V∖S (resp., in V) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the k-tuple dominating set problem (this k is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respect to k, where k is the size of solution.

Power domination of the cartesian product of graphs

In this paper, we first give a brief survey on the power domination of the Cartesian product of graphs. Then we conjecture a Vizing-like inequality for the power domination problem, and prove that the inequality holds when at least one of the two graphs is a tree.

Weighted restrained domination in subclasses of planar graphs

An undirected simple and connected graph is denoted by G(V,E), where V and E are the vertex-set and the edge-set of G, respectively. For any subset S of V,〈S〉 denotes the subgraph of G induced by S. A subset Q of V is a restrained dominating set of G if ⋃u∈QN[u]=V and no isolated vertices appear in 〈V−Q〉, where N(x)={u|(u,x)∈E} and N[x]=N(x)∪{x}, for all x∈V. This paper studies the Weighted Restrained Domination problem (the WRD problem) on graphs G(V,E,w), where w is a function giving a positive weight w(v) to each vertex v. The problem is to find a restrained dominating set D of the given graph G(V,E,w) and the objective is to minimize w(D)=∑v∈Dw(v). When w(v)=1, for all v∈V, the WRD problem is abbreviated as the RD problem. The first result shows that the WRD problem on cactus graphs can be solved in linear time. The second result proves that the RD problem on planar bipartite graphs remains NP-hard.

Domination by positive Dunford–Pettis completely continuous operators

We give a necessary and sufficient conditions for which the domination problem admits a positive solution for the class of positive Dunford–Pettis completely continuous operators. Also, we prove that if \(S : E \longrightarrow E\) is dominated by a Dunford–Pettis completely continuous operator, then the second power operator \(S^{2}\) is also Dunford–Pettis completely continuous.

Schur operators and domination problem

In this paper we are concerned with developing generalizing concepts of Dunford–Pettis operators analogous to the generalization of compact operators by strictly singular operators. Also, we give some new results concerning the domination problem in the setting of positive operators between Banach lattices.