Domination Problem In Graphs Research Articles

Power domination in Mycielskians of n-spiders

The power domination problem in graphs was introduced to model the monitoring problem in electric networks. It is to find a set of vertices, S ⊆ V ( G ) , of a graph G, called the power dominating set, that monitors V(G) by application of two rules, namely, domination and propagation. In this article, we discuss the power domination problem in the Mycielskians of n-spiders. We show that the power domination number of the Mycielskians of n-spiders is bounded above by n and also characterize the Mycielskians of n-spiders with γ P = 1 and γ P = n .

Algorithms and hardness results for edge total domination problem in graphs

For a graph G=(V,E) without an isolated edge, a set D⊆E is an edge total dominating set of G if every edge e∈E is adjacent to at least one edge of D. The Minimum Edge Total Dominating Set problem is to compute a minimum cardinality edge total dominating set of G. It is known that the decision version of the problem is NP-complete for bipartite graphs, chordal graphs, and planar graphs. On the positive side, the problem is efficiently solvable for trees. In this paper, we further study the complexity of this problem in other graph classes. We design a linear-time algorithm to compute a minimum cardinality edge total dominating set in chain graphs, a subclass of bipartite graphs. We also propose linear-time algorithms for two subclasses of chordal graphs, namely, split graphs and proper interval graphs with no cut vertices. In addition, we show that the problem is APX-complete for graphs with maximum degree 3 and propose an approximation algorithm for the problem in k-regular graphs, where k≥4. We discuss the complexity difference between the Minimum Edge Dominating Set problem and Minimum Edge Total Dominating Set problem which seem to be closely related.

Power domination in Mycielskian of spiders

The power domination problem in graphs consists of finding a minimum set of vertices that monitors the entire graph G governed by two ‘monitoring rules’- domination and propagation. A set is a power dominating set (PDS) if it can monitor all vertices of G. The minimum cardinality of a PDS of G is called the power domination number, , of G. In this paper, we study the power domination problem in Mycielskian of spiders. For a spider T, we have and . We characterize spiders, T, for which and

Dual domination problems in graphs

We introduce a novel domination problem, which we call Dual Domination (DD). We assume that the nodes in a network are partitioned into two categories: Positive nodes (V+) and negative nodes (V−). We study the Maximum Bounded Dual Domination, where, given a bound k, the problem is to find a set D⊆V+, which maximizes the number of nodes dominated in V+, dominating at most k nodes in V−. We show that such problem is hard to approximate to a factor better than (1−1/e). We give a polynomial time algorithm with approximation guaranteed (1−e−1/Δ), where Δ represents the maximum number of neighbors in V+ of any node in V−, and an O(|V|k2) time algorithm to solve the problem on trees. We also study two related problems named Maximum Dual Domination and minimum Negative Dual Domination. For both problems, we show hardness results and provide O(|V|) time algorithms on trees.

Complexity of signed total $k$-Roman domination problem in graphs

Let $G$ be a simple graph with finite vertex set $V(G)$ and $S = \{-1, 1, 2\}$. A signed total Roman $k$-dominating function (STRkDF) on a graph $G$ is a function $f:V(G)\to S$ such that (i) any vertex $y$ with $f(y) = -1$ is adjacent to at least one vertex $t$ with $f(t) = 2, $ (ii) $\sum_{t\in N(y)}f(t)\geq k$ holds for any vertex $y$. The $weight$ of an STRkDF $f$, denoted by $\omega(f)$, is $\sum_{y\in V(G)}f(y)$, and the minimum weight of an STRkDF is the <i>signed total Roman k-domination number</i>, $\gamma_{stR}^k(G), $ of $G$. In this article, we prove that the decision problem for the signed total Roman $k$-domination is NP-complete on bipartite and chordal graphs for $k\in\{1, 2\}$.

Roman \{2\}-domination problem in graphs

Roman <i>\{2\}</i>-domination problem in graphs

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.

The complexity of secure domination problem in graphs

The complexity of secure domination problem in graphs

More results on the complexity of domination problems in graphs

Given a graph G = (V, E) and an integer r &ge; 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 &sub; 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 $n \geq m \geq 8$.

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.

A Dynamic Domination Problem

A dynamic domination problem in graphs is studied in which an innite sequence of attacks occurs at vertices with guards and the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex. 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. We are concerned with the minimum number of guards required to achieve this goal against any arbitrary sequence of attacks; this number is called the swap number. The swap number lies between the domination and independence numbers of the graph; bounds for some classes of graphs are examined in this paper.

Complexity of total outer-connected domination problem in graphs

A set D⊆V is called a total dominating set of a graph G=(V,E), if for all v∈V, |NG(v)∩D|≥1. A total dominating set D of G=(V,E) is called a total outer-connected dominating set if G[V∖D] is connected. The Minimum Total Outer-connected Domination problem for a graph G=(V,E) is to find a total outer-connected dominating set of minimum cardinality. The Total Outer-connected Domination Decision problem, the decision version of the Minimum Total Outer-connected Domination problem, is known to be NP-complete for general graphs. In this paper, we strengthen this NP-completeness result by showing that the Total Outer-connected Domination Decision problem remains NP-complete for chordal graphs, split graphs, and doubly chordal graphs. We prove that the Total Outer-connected Domination Decision problem can be solved in linear time for bounded tree-width graphs. We, then, propose a linear time algorithm for computing the total outer-connected domination number, the cardinality of a minimum total outer-connected dominating set, of a tree. We prove that the Minimum Total Outer-connected Domination problem cannot be approximated within a factor of (1−ε)ln|V| for any ε>0, unless NP ⊆ DTIME (|V|O(loglog|V|)). Finally, we show that the Minimum Total Outer-connected Domination problem is APX-complete for bounded degree graphs.

Downhill domination problem in graphs

A path π=(v1,v2,⋯,vk+1) in a graph G=(V,E) is a downhill path if for every i, 1≤i≤k, deg(vi)≥deg(vi+1), where deg(vi) denotes the degree of vertex vi∈V. A downhill dominating set DDS is a set S⊆V having the property that every vertex v∈V lies on a downhill path originating from some vertex in S. The downhill domination numberγdn(G) equals the minimum cardinality of a DDS of G. A DDS having minimum cardinality is called a γdn-set of G. In this note, we give an enumeration of all the distinct γdn-sets of G, and we show that there is a linear time algorithm to determine the downhill domination number of a graph.

Hardness results, approximation and exact algorithms for liar's domination problem in graphs

A subset L⊆V of a graph G=(V,E) is called a liar's dominating set of G if (i) |NG[u]∩L|≥2 for every vertex u∈V, and (ii) |(NG[u]∪NG[v])∩L|≥3 for every pair of distinct vertices u,v∈V. The Min Liar Dom Set problem is to find a liar's dominating set of minimum cardinality of a given graph G and the Decide Liar Dom Set problem is the decision version of the Min Liar Dom Set problem. The Decide Liar Dom Set problem is known to be NP-complete for general graphs. In this paper, we first present approximation algorithms and hardness of approximation results of the Min Liar Dom Set problem in general graphs, bounded degree graphs, and p-claw free graphs. We then show that the Decide Liar Dom Set problem is NP-complete for doubly chordal graphs and propose a linear time algorithm for computing a minimum liar's dominating set in block graphs.

Generalized limited packings of some graphs with a limited number of [formula omitted]-partners

By using modular decomposition and handling certain graph operations such as join and union, we show that the Generalized Limited Packing Problem—NP-complete in general—can be solved in polynomial time in some graph classes with a limited number of P4-partners; specifically P4-tidy graphs, which contain cographs and P4-sparse graphs. In particular, we describe an algorithm to compute the associated numbers in polynomial time within these graph classes. In this way, we generalize some of the previous results on the subject. We also make some progress on the study of the computational complexity of the Generalized Multiple Domination Problem in graphs.

Limited Packing and Multiple Domination problems: Polynomial time reductions

The Limited Packing and Multiple Domination problems in graphs have closely-related definitions and the same computational complexity on several graph classes. In this work we present two polynomial time reductions between them. Besides, we take into consideration generalized versions of these problems and obtain polynomial time reductions between each one and its generalized version.

Algorithmic aspects of the [formula omitted]-domination problem in graphs

For a positive integer k, a k-dominating set of a graph G is a subset D⊆V(G) such that every vertex not in D is adjacent to at least k vertices in D. The k-domination problem is to determine a minimum k-dominating set of G. This paper studies the k-domination problem in graphs from an algorithmic point of view. In particular, we present a linear-time algorithm for the k-domination problem for graphs in which each block is a clique, a cycle or a complete bipartite graph. This class of graphs includes trees, block graphs, cacti and block-cactus graphs. We also establish NP-completeness of the k-domination problem in split graphs.

On the mixed domination problem in graphs

A mixed dominating set of a simple graph G=(V,E) is a subset D⊆V∪E such that every vertex or edge not in D is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to determine a minimum mixed dominating set of G. This paper studies mixed domination in graphs from an algorithmic point of view. In particular, a linear-time labeling algorithm for the mixed domination problem in cacti is presented. In addition, we fix an incomplete proof of the NP-completeness of the mixed domination problem in split graphs Zhao et al. (2011) [23]. Finally, we establish a primal–dual algorithm for the mixed domination problem in trees.