Weak Domination Research Articles

Notes on the value function approach to multiobjective bilevel optimization

This paper is concerned with the value function approach to multiobjective bilevel optimization which exploits a lower-level frontier-type mapping in order to replace the hierarchical model of two interdependent multiobjective optimization problems by a single-level multiobjective optimization problem. As a starting point, different value-function-type reformulations are suggested and their relations are discussed. Here, we focus on the situations where the lower-level problem is solved up to efficiency or weak efficiency, and an intermediate solution concept is suggested as well. We study the graph-closedness of the associated efficiency-type and frontier-type mappings. These findings are then used for two purposes. First, we investigate existence results in multiobjective bilevel optimization. Second, for the derivation of necessary optimality conditions via the value function approach, it is inherent to differentiate frontier-type mappings in a generalized way. Here, we are concerned with the computation of upper coderivative estimates for the frontier-type mapping associated with the setting where the lower-level problem is solved up to weak efficiency. We proceed in two ways, relying, on the one hand, on a weak domination property and, on the other hand, on a scalarization approach. Illustrative examples visualize our findings and some flaws in the related literature.

Weak and strong domination on some graphs

Let G = (V(G), E(G)) be a graph and uvεE. A subset D ⊆ V of vertices is a dominating set if every vertex in V − D is adjacent to at least one vertex of D. The domination number is the minimum cardinality of a dominating set. Let u and v be elements of V. Then, u strongly dominates u and v weakly dominates u if (i)uvεE and (ii)deg(u) ≥ deg(v). A set D ⊆ V is a strong (weak) dominating set (sd-set)(wd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. The strong (weak) domination number γs(γw) of G is the minimum cardinality of a sd-set (wd-set). In this paper, the strong and weak domination numbers of comet, double comet, double star and theta graphs are given. The theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, MST construction and real-time animation.

The convex and weak convex domination number of convex polytopes

The convex and weak convex domination number of convex polytopes

A novel concept of domination in m-polar interval-valued fuzzy graph and its application.

The concept of domination is one of the most significant topics in graph theory to handle unpredictable phenomena. In this study, an unprecedented idea of domination is introduced in m-polar interval-valued fuzzy graph (m-PIVFG). Domination number (DN), isolated vertex, total dominating set, independent set of domination on m-PIVFG are discussed. Some algebraic properties of domination on m-PIVFG are investigated. Weak domination, strong domination, split and non-split domination, cototal and global dominating sets on m-PIVFG are investigated with some fundamental hypotheses and models. We explore the concept of domination in m-PIVFG by solving a case study of locating new facilities to handle a catastrophe reaction activity due to the “COVID-19 pandemic” in West Bengal, India. Ultimately, conclusions and avenues of future scopes are placed at the end of this study.

Stability properties of the core in a generalized assignment problem

We show that the core of a generalized assignment problem satisfies two types of stability properties. First, the core is the unique stable set defined using the weak domination relation when outcomes are restricted to individually rational and pairwise feasible ones. Second, the core is the unique stable set with respect to a sequential domination relation that is defined by a sequence of weak domination relations that satisfy outsider independence. An equivalent way of stating this result is that the core satisfies the property commonly stated as the existence of a path to stability. These results add to the importance of the core in an assignment problem where agents' preferences may not be quasilinear.

Weak Roman subdivision number of graphs

A Roman dominating function (RDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex u for which [Formula: see text] is adjacent to at least one vertex v for which [Formula: see text]. A vertex [Formula: see text] with [Formula: see text] is undefended if it is not adjacent to a vertex with [Formula: see text]. The function [Formula: see text] is a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text] such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] if [Formula: see text] has no undefended vertex. The Roman domination subdivision number of a graph [Formula: see text] is the minimum number of edges that must be subdivided in order to increase the Roman domination number of [Formula: see text]. We introduce the concept of weak Roman subdivision number of a graph [Formula: see text], denoted by [Formula: see text] as the minimum number of edges that must be subdivided in order to increase the weak Roman domination number of [Formula: see text]. In this paper, we determine the exact values of the weak Roman subdivision number for paths, cycles and complete bipartite graphs. We obtain bounds for the weak Roman subdivision number of a graph and characterize the extremal graphs.

Stability Properties of the Core in a Generalized Assignment Problem

We show that the core of a generalized assignment problem satisfies two types of stability properties. First, the core is the unique stable set defined using the weak domination relation when outcomes are restricted to individually rational and pairwise feasible ones. Second, the core is the unique stable set with respect to a sequential domination relation that is defined by a sequence of weak domination relations that satisfy outsider independence. An equivalent way of stating this result is that the core satisfies the property commonly stated as the existence of a path to stability. These results add to the importance of the core in an assignment problem where agents' preferences may not be quasilinear.

Novel Properties for Total Strong - Weak Domination Over Bipolar Intuitionistic Fuzzy Graphs

Through this research study, we introduced and discussed total strong (weak) domination concept of bipolar intuitionistic fuzzy graphs and in define strong domination bipolar intuitionistic fuzzy graph also strong domination. Theorems, examples and some properties of these concept are discussed.

An algorithm to recognize weak roman domination stable trees under vertex deletion

Let [Formula: see text] be a graph and [Formula: see text] be a function. A vertex [Formula: see text] with weight [Formula: see text] is said to be undefended with respect to [Formula: see text], if it is not adjacent to any vertex with positive weight. The function [Formula: see text] is a weak Roman dominating function (WRDF) if each vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text] such that the function [Formula: see text] defined by [Formula: see text], [Formula: see text] and [Formula: see text] if [Formula: see text], has no undefended vertex. The weight of [Formula: see text] is [Formula: see text]. The weak Roman domination number, denoted by [Formula: see text], is the minimum weight of a WRDF on [Formula: see text]. In this paper, we present two linear time algorithms one that obtains the weak Roman domination number of an arbitrary tree and the labeling of its vertices, to produce the weak Roman domination number, and the other, that determines whether the given tree is in [Formula: see text].

Independent strong weak domination: A mathematical programming approach

Let [Formula: see text] be a graph. A subset [Formula: see text] of vertices is a dominating set if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination number is the minimum cardinality of a dominating set. Let [Formula: see text]. Then, [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a strong (weak) dominating set of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number of [Formula: see text] is the minimum cardinality of a strong (weak) dominating set. A set [Formula: see text] is an independent (or stable) set if no two vertices of [Formula: see text] are adjacent. The independent domination number of [Formula: see text] (independent strong domination number, independent weak domination number, respectively) is the minimum size of an independent dominating set (independent strong dominating set, independent weak dominating set, respectively) of [Formula: see text]. In this paper, mathematical models are developed for the problems of independent domination and independent strong (weak) domination of a graph. Then test problems are solved by the GAMS software, the optima and execution times are implemented. To the best of our knowledge, this is the first mathematical programming formulations for the problems, and computational results show that the proposed models are capable of finding optimal solutions within a reasonable amount of time.

Domination in generalized unit and unitary Cayley graphs of finite rings

Let R be a finite commutative ring with nonzero identity and U(R) be the set of all units of R. The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U(R) such that x + uy is a unit in R. Also, $$\overline{\Gamma}$$ denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as $$\overline{\Gamma}$$ and characterize all γ-sets in Γ and $$\overline{\Gamma}$$. Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of $$\overline{\Gamma}$$.

Weak Domination in LICT Graphs

Weak Domination in LICT Graphs

Minimal Enforceability and Indirect Domination Relations in the Shapley-Scarf Economy

We consider the von Neumann-Morgenstern (vNM) stable sets based on a farsighted version of weak domination relation for the barter model with indivisible goods of Shapley and Scarf (1974). Recent research on farsighted stable sets has focused on coalitional sovereignty, a term used by Ray and Vohra (2015) regarding what coalitions can do to other coalitions, and they note the importance of this issue and how they may affect the results in characteristic function form games. However, defining a farsighted version of indirect weak domination introduces an additional issue of what coalitions can or cannot do from the inside, an issue which needed not be analyzed for the stronger versions of indirect domination nor for myopic domination relations in the literature. We first impose a minimality condition on the coalitions that can deviate in just one step, as coalitions in this model represent trading cycles, where such a minimality condition is often imposed. Under the domain of preferences defined by Klaus, Klijn, and Walzl (2010), each Pareto efficient allocation in the core and its Pareto indifferent allocations constitute a vNM stable set defined by a farsighted weak domination that respects the minimal enforceability condition, and those are the only essentially singleton such vNM stable sets.

Total Weak Roman Domination in Graphs

Given a graph G = ( V , E ) , a function f : V → { 0 , 1 , 2 , ⋯ } is said to be a total dominating function if ∑ u ∈ N ( v ) f ( u ) > 0 for every v ∈ V , where N ( v ) denotes the open neighbourhood of v. Let V i = { x ∈ V : f ( x ) = i } . We say that a function f : V → { 0 , 1 , 2 } is a total weak Roman dominating function if f is a total dominating function and for every vertex v ∈ V 0 there exists u ∈ N ( v ) ∩ ( V 1 ∪ V 2 ) such that the function f ′ , defined by f ′ ( v ) = 1 , f ′ ( u ) = f ( u ) − 1 and f ′ ( x ) = f ( x ) whenever x ∈ V \ { u , v } , is a total dominating function as well. The weight of a function f is defined to be w ( f ) = ∑ v ∈ V f ( v ) . In this article, we introduce the study of the total weak Roman domination number of a graph G, denoted by γ t r ( G ) , which is defined to be the minimum weight among all total weak Roman dominating functions on G. We show the close relationship that exists between this novel parameter and other domination parameters of a graph. Furthermore, we obtain general bounds on γ t r ( G ) and, for some particular families of graphs, we obtain closed formulae. Finally, we show that the problem of computing the total weak Roman domination number of a graph is NP-hard.

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.

Weak dominating sets and Weak Domination Polynomial of Complete Graphs

Weak dominating sets and Weak Domination Polynomial of Complete Graphs

Strong and weak cobondage numbers of a graph

Let G = (V, E) be a graph. If uv ∈ E, we say that u and v dominate each other. If uv ∈ E and deg u ≥ deg v, we say that u strongly dominates v and v weakly dominates u. A subset D of V is called a strong dominating set of G if every vertex v ∈ V – D is strongly dominated by some u ∈ D. The strong domination number γs(G) of G is the size of its smallest strong dominating set. Similarly, a weak dominating set and the weak domination number γw(G) of G are defined. We define the strong cobondage number bsc(G) of G to be the minimum number of edges to be added to G to decrease the strong domination number γs(G). Similarly, we define weak cobondage number bwc(G) of G. In this paper, strong and weak cobondage numbers of some standard graphs are found. Upper bounds for bsc(G) and bwc(G) for trees are obtained; also graphs attaining these bounds are characterised. Upper bounds for bsc(G) in terms of D(G) and n are given and graphs that attain these bounds are established. An upper bound for bwc is determined for any graph G. An upper bound for bsc(G) + γs(G) is found and graphs that attain these bounds are characterised.

The entropy of 𝐶¹-diffeomorphisms without a dominated splitting

A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local C 1 C^1 -perturbations. Our main theorem shows that when one works in the C 1 C^1 -topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound following from Ruelle’s inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller). This optimal entropy creation yields a number of consequences for C 1 C^1 -generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism, and prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero.

STRONG AND WEAK DOMINATION OF MIDDLE GRAPHS OF PATH AND CYCLE GRAPHS

Let G =(V,E) be a simple, finite, undirected and connected graph. Let then strongly dominates v if (i) and (ii) .A non empty subset is a strong dominating set (sd-set) of G if every vertex in is strongly dominated by at least one vertex in S. Similarly, a weak dominating set (wd-set) is defined. The strong (weak)domination ᵞs (ᵞw) number of G is the minimum cardinality of a sd-set (wdset). The middle graph M(G) of a graph is obtained by subdividing each edge of G exactly once and joining all these newly added middle vertices to the adjacent edges of G. Let denote the greatest integer not greater than and denote the least integer not less than . In this paper, we investigate the strong and weak domination number of the middle graphs of the path Pn and the cycle Cn . AMS Subject Classification: 05C09.

Improved integer linear programming formulation for weak Roman domination problem

Let \(f:V \rightarrow \{0,1,2\}\) be a function, \(G=(V,E)\) be a graph with a vertex set V and a set of edges E and let the weight of the vertex \(u \in V\) be defined by f(u). A vertex u with property \(f(u)=0\) is considered to be defended with respect to the function f if it is adjacent to a vertex with positive weight. Further, the function f is called a weak Roman dominating function (WRDF) if for every vertex u with property \(f(u)=0\) there exists at least one adjacent vertex v with positive weight such that the function \(f':V \rightarrow \{0,1,2\}\) defined by \(f'(u)=1\), \(f'(v)=f(v)-1\) and \(f'(w)=f(w)\), \(w \in V \setminus \{u,v\}\) has no undefended vertices. In this paper, an optimization problem of finding the WRDF f such that \(\sum _{u \in V}{f(u)}\) is minimal, known as the weak Roman domination problem (WRDP), is considered. Therefore, a new integer linear programing (ILP) formulation is proposed and compared with the one known from the literature. Comparison between the new and the existing formulation is made through computational experiments on a grid, planar, net and randomly generated graphs known from the literature and up to 600 vertices. Tests were run using standard CPLEX and Gurobi optimization solvers. The obtained results demonstrate that the proposed new ILP formulation clearly outperforms the existing formulation in the sense of solutions’ quality and running times.