Weak Domination Research Articles

On parameters related to strong and weak domination in graphs

Let G be a graph. Then μ( G)⩽| V( G)|− δ( G) where μ( G) denotes the weak or independent weak domination number of G and μ( G)⩽| V( G)|− Δ( G) where μ( G) denotes the strong or independent strong domination number of G. We give necessary and sufficient conditions for equality to hold in each case and also describe specific classes of graphs for which equality holds. Finally, we show that the problems of computing i w and i st are NP-hard, even for bipartite graphs.

The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences

The No Show Paradox (there is a voter who would rather not vote) is known to affect every Condorcet voting function. This paper analyses two strong versions of this paradox in the context of Condorcet voting correspondences. The first says that there is a voter whose favorite candidate loses the election if she votes honestly, but gets elected if she abstains. The second says that there is a voter whose least preferred candidate gets elected if she votes honestly, but loses the election if she abstains. All Condorcet correspondences satisfying some weak domination properties are shown to be affected by these strong forms of the paradox. On the other hand, with the exception of the Simpson-Cramer Minmax and the Young rule, all the Condorcet correspondences that (to the best of our knowledge) are proposed in the literature suffer from these two paradoxes.

Strong weak domination and domination balance in a graph

Let G = ( V, E) be a graph and u, v ∈ V. Then, u strongly dominates v and v weakly dominates u if (i) uv ∈ E and (ii) deg u ⩾ deg v. A set D ⊂ V is a strong-dominating set (sd-set) of G if every vertex in V − D is strongly dominated by at least one vertex in D. Similarly, a weak-dominating set (wd-set) is defined. The strong (weak) domination number γ s ( γ w) of G is the minimum cardinality of an sd-set (wd-set). Besides investigating some relationship of γ s and γ w with other known parameters of G, some bounds are obtained. A graph G is domination balanced if there exists an sd-set D 1 and a wd-set D 2 such that D 1 ∩ D 2 = 0. A study of domination balanced graphs is initiated.

Stable Outcomes in Discrete and Continuous Models of Two-Sided Matching: a Unified Treatment

We present a unified treatment of a class of two-sided matching models that includes discrete models (such as the marriage model of Gale and Shapley) and continuous models (such as the assignmnent model of Shapley aud Shubik and the generalized assignment model of Demange aud Gale). In contrast with previous, treatments, the parallel conclusions for the two sets of models are derived here in the same way from the same assumptions. We show that the results ill question all follow closely from the assumptions that the core coincides with the core defined by weak domination. In the marriage model, the assumption of strict preferences causes these two sets to coincide, while in the continuous models the two sets coincide because agents have continuous preferences and prices can be adjusted continuously.

The Complexity of Eliminating Dominated Strategies

This paper deals with the computational complexity of some yes/no problems associated with sequential elimination of strategies using three domination relations: strong domination (strict inequalities), weak domination (weak inequalities), and domination (the asymmetric part of weak domination). Classification of various problems as polynomial or NP-complete seems to suggest that strong domination is a simple notion, whereas weak domination and domination are complicated ones.

Kernels and Choquet capacities

M. Brelot showed that the capacity corresponding to a function-kernel is a Choquet capacity, provided that the kernel satisfies the principle of equilibrium, the weak domination principle and the adjoint kernel satisfies the weak principle of equilibrium. This result is not applicable for a series of important kernels in potential theory (e.g. the fundamental solution of the heat equation, or the Kolmogorov equation), since the above principles no longer hold in this situation. New principles for function kernels guaranteeing that the capacity is a Choquet capacity are introduced and applied in the framework of balayage spaces. In particular, polar and adjoint polar sets are shown to coincide in this context.

Some Properties of Weak Domination in an Exchange Market with Indivisible Goods

Some Properties of Weak Domination in an Exchange Market with Indivisible Goods

The College Admissions Problem Revisited

The college admissions problem is perhaps the simplest model of many-to-one matching in two-sided markets, such as labor markets. The authors show that the set of stable outcomes (which is equal to the core defined by weak domination) has some surprising properties not found in models of one-to-one matching. These properties may help to explain the success that this kind of model has had in explaining empirical observations. Copyright 1989 by The Econometric Society.

The Knowledge Assumption in the Theory of Strategic Voting

individual may indeed lead him to reject his sincere strategy. In the present paper, the same class of voting procedures is analyzed, and a similar result is shown to hold in terms of a new concept, that of weak domination in the extended sense between the strategies of an individual. The basic idea of the extension is to account for situations where only partial information on other individuals' preferences is held by members of the society, and to show that this partial would be sufficient for a rational individual to choose nonsincere strategies under certain conditions. The main motivation of this note is thus to relax the rather restrictive assumption of perfect knowledge underlying most of the analysis in the literature on strategic voting, and to examine the problem of strategic voting with a weaker assumption.

Weak versus strong domination in a market with indivisible goods

The core of a market in indivisible goods can be defined in terms of strong domination or weak domination. The core defined by strong domination is always non-empty, but may contain points which are unstable in a dynamic sense. However, it is shown that there are always stable points in the core, and a characterization is obtained. The core defined by weak domination is always non-empty when there is no indifference, and has no instability problems. In this case, the core coincides with the unique competitive allocation.

Pivotal measures in the case of weak domination

Pivotal measures in the case of weak domination

Supplement to ``Weak domination principle''

Supplement to ``Weak domination principle''

Weak domination principle

Weak domination principle