2-dominating Set Research Articles

Spanning trees with disjoint 2 dominating and 2-dominating sets

Spanning trees with disjoint 2 dominating and 2-dominating sets

Generalizing Erdős, Moon and Moser's result — The number of k-dominating independent sets

We study the existence and the number of k-dominating independent sets in certain graph families. A set D is k-dominating if each vertex in V(G)\\D is adjacent to at least k vertices of D. While the case k=1 namely the case of maximal independent sets – which is originated from Erdős and Moser – is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k-dominating independent sets in n-vertex graphs is between ck⋅2n2k and ck′⋅2nk+1 if k≥2, moreover the maximum number of 2-dominating independent sets in n-vertex graphs is between c⋅1.22n and c′⋅1.246n. Graph constructions containing a large number of k-dominating independent sets are coming from product graphs, complete bipartite graphs and finite geometries. The product graph construction is associated with the number of certain MDS codes.

Bounds on the [formula omitted]-domination number

In a graph G, a set D⊆V(G) is called 2-dominating set if each vertex not in D has at least two neighbors in D. The 2-domination number γ2(G) is the minimum cardinality of such a set D. We give a method for the construction of 2-dominating sets, which also yields upper bounds on the 2-domination number in terms of the number of vertices, if the minimum degree δ(G) is fixed. These improve the best earlier bounds for any 6≤δ(G)≤21. In particular, we prove that γ2(G) is strictly smaller than n∕2, if δ(G)≥6. Our proof technique uses a weight-assignment to the vertices where the weights are changed during the procedure.

Distance 2-domination in prisms of graphs

A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ? ( V ( G ) - D ) and D is at most two. Let ? 2 ( G ) denote the size of a smallest distance 2 -dominating set of G . For any permutation p of the vertex set of G , the prism of G with respect to p is the graph pG obtained from G and a copy G ' of G by joining u ? V ( G ) with v ' ? V ( G ' ) if and only if v ' = p ( u ) . If ? 2 ( pG ) = ? 2 ( G ) for any permutation p of V ( G ) , then G is called a universal ? 2 - fixer. In this work we characterize the cycles and paths that are universal ? 2 -fixers.

Improved distributed local approximation algorithm for minimum 2-dominating set in planar graphs

In this paper we consider the 2-dominating set problem (2MDS). We look for a smallest subset of vertices D⊆V with the property that every vertex in V∖D is adjacent to at least 2 vertices of D. We are interested in the distributed complexity of this problem in the local model, where the nodes have no identifiers but there is a port ordering available. We propose a distributed local (constant time) algorithm yielding a 6-approximation in the class of planar graphs. Earlier result shows that in this case, for any ϵ>0, there is no deterministic distributed local/constant-round algorithm providing a (5−ϵ)-approximation of the 2MDS.

Trees with 2-Reinforcement Number Three

A vertex subset S of a graph G is a 2-dominating set of G if every vertex not in S is adjacent to two vertices of S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-reinforcement number r2(G) is the smallest number of extra edges whose addition to G results in a graph Gwith γ2(G � )<γ 2(G) .L etT be a tree. It is showed by Lu, Hu, and Xu that r2(T ) ≤ 3. In this paper, we will show that r2(T ) = 3 if and only if there is a 2-dominating set S of T such that T contains neither S-vulnerable vertices nor S-vulnerable paths. Keywords 2-Domination · 2-Reinforcement number · S-vulnerable · Trees

On trees with equal 2-domination and 2-outer-independent domination numbers

For a graph G = (V, E), a subset D ⊆ V(G) is a 2-dominating set if every vertex of V(G)\D has at least two neighbors in D, while it is a 2-outer-independent dominating set if additionally the set V(G)\D is independent. The 2-domination (2-outer-independent domination, respectively) number of G, is the minimum cardinality of a 2-dominating (2-outer-independent dominating, respectively) set of G. We characterize all trees with equal 2-domination and 2-outer-independent domination numbers.

On Constructing Minimum 2-Connected 2-Domiating Set with Maximum Spectrums in Cognitive Radio Networks

Abstract Connected dominating set (CDS) is a representative technique for constructing virtual backbones of wireless networks to improve the efficiency of communication. Most of existing works on CDS aim to construct the minimum CDS (MCDS). However, MCDS may not be sufficient in cognitive radio networks where communication links are prone to failure due to random activities of primary users. To solve the problem, an algorithm is proposed in this paper to construct a CDS with redundancy and the maximum spectrums in CRNs. Moreover, with an integer programming formulation, an algorithm is designed to obtain the minimum 2-conneted 2-dominating set with maximum spectrums (MSs) in CRNs. To the best of our knowledge, this is the first integer programming formulation for the problem and the concept of MSs is put forward first. Simulation results indicate that our algorithm performs well.

Relating the annihilation number and the 2-domination number of a tree

A set S of vertices in a graph G is a 2-dominating set if every vertex of G not in S is adjacent to at least two vertices in S. The 2-domination numberγ2(G) is the minimum cardinality of a 2-dominating set in G. The annihilation numbera(G) is the largest integer k such that the sum of the first k terms of the nondecreasing degree sequence of G is at most the number of edges in G. The conjecture-generating computer program, Graffiti.pc, conjectured that γ2(G)≤a(G)+1 holds for every connected graph G. It is known that this conjecture is true when the minimum degree is at least 3. The conjecture remains unresolved for minimum degree 1 or 2. In this paper, we prove that the conjecture is indeed true when G is a tree, and we characterize the trees that achieve equality in the bound. It is known that if T is a tree on n vertices with n1 vertices of degree 1, then γ2(T)≤(n+n1)/2. As a consequence of our characterization, we also characterize trees T that achieve equality in this bound.

On Constructing 2-Connected2-Dominating Set Using Distributed Algorithm

This paper proposes a distributed algorithm to construct a 2-connected 2-dominating set. The main ingredient of this algorithm is that the nodes in the loop are 2-vertex connected. It starts from any node, and forms a localized loop made by dominating nodes. The nodes in the loop are 2-vertex connected. Basing on this loop, the algorithm continue to construct other loops formed by dominating nodes until the nodes out loops are 2-dominated.

The 2-dominating set polytope of cycles and related graph classes

Given a graph G and a nonnegative integer number k, a k-dominating set in G is a subset of vertices D such that every vertex in the graph is adjacent to at least k elements of D. The k-dominating set polytope is the convex hull of the incidence vectors of k-dominating sets in G. This is a natural generalization of the well-known dominating set polytope in graphs. In this work we present a complete description of the 2-dominating set polytope of cycles and show that every facet of this polytope can be separated in polynomial time. We use our findings to derive facets of the 2-dominating set polytope of cacti, i.e. graphs obtained as 1-sums of cycles and edges.

On the ratio between 2-domination and total outer-independent domination numbers of trees

A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V (G) | D has at least two neighbors in D. A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) | D is independent. The 2-domination (total outer-independent domination, respectively) number of a graph G is the minimum cardinality of a 2-dominating (total outer-independent dominating, respectively) set of G. We investigate the ratio between 2-domination and total outer-independent domination numbers of trees.

Minimal 2-dominating sets in trees

We provide an algorithm for listing all minimal 2-dominating sets of a tree of ordernin time 𝒪(1.3248n). This implies that every tree has at most 1.3248nminimal 2-dominating sets. We also show that this bound is tight.

2-bondage in graphs

A 2-dominating set of a graph G=(V, E) is a set D of vertices of G such that every vertex of V(G) ∖ D has at least two neighbours in D. The 2-domination number of a graph G, denoted by γ2(G), is the minimum cardinality of a 2-dominating set of G. The 2-bondage number of G, denoted by b 2(G), is the minimum cardinality among all sets of edges E′ ⊆ E such that γ2(G−E′)>γ2(G). If for every E′ ⊆ E we have γ2(G−E′)=γ2(G), then we define b 2(G)=0, and we say that G is a γ2-strongly stable graph. First, we discuss the basic properties of 2-bondage in graphs. We find the 2-bondage numbers for several classes of graphs. Next we show that for every non-negative integer there exists a tree with such 2-bondage number. Finally, we characterize all trees with 2-bondage number equaling one or two.

On graphs with disjoint dominating and 2-dominating sets

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the graph.

Restrained 2-Domination Number of Complete Grid Graphs

In this paper we present 2-dominating sets for complete grid graphs $G_{k,n}$ by using V-merge and H-merge techniques recursively and then find 2-domination and restrined 2-domination numbers of complete grid graphs. We also verify the Vizing’s conjecture for both domination parameters.

On trees with double domination number equal to the 2-outer-independent domination number plus one

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G, such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2-dominating set if every vertex of V (G) \ D has at least two neighbors in D, while it is a 2-outer-independent dominating set of G if additionally the set V (G)\D is independent. The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G. This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.

Trees with equal 2-domination and 2-independence numbers

Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number 2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number �2(G). Fink and Jacobson proved that 2(G) � �2(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.

On graphs with disjoint dominating and 2-dominating sets

A D D 2 -pair of a graph G is a pair ( D , D 2 ) of disjoint sets of vertices of G such that D is a dominating set and D 2 is a 2 -dominating set of G . Although there are infinitely many graphs that do not contain a D D 2 -pair, we show that every graph with minimum degree at least 2 has a D D 2 -pair. We provide a constructive characterization of trees that have a D D 2 -pair and show that K 3 , 3 is the only connected graph with minimum degree at least 3 for which D ∪ D 2 necessarily contains all vertices of the graph.

Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

In this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We transform the problem of monitoring a piecewise-convex polygon to the problem of 2- dominating a properly defined triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in its 2-dominating set. We show that: (1) ⌊ n + 1 3 ⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊ 2 n + 1 5 ⌋ edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ⌊ n + 1 3 ⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊ 2 n + 1 5 ⌋ in O ( n 2 ) time and O ( n ) space, and (3) an edge 2-dominating set of size ⌊ 3 n 7 ⌋ in O ( n ) time and space. Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ⌊ n + 1 3 ⌋ in O ( n log n ) time, (2) an edge guard set of size ⌊ 2 n + 1 5 ⌋ in O ( n 2 ) time, and (3) an edge guard set of size ⌊ 3 n 7 ⌋ in O ( n log n ) time. All space requirements are linear. Finally, we show that ⌊ n 3 ⌋ mobile or ⌈ n 3 ⌉ edge guards are sometimes necessary. When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈ n + 1 4 ⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈ n + 1 4 ⌉ , can be computed in O ( n ) time and space.