Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open $k$-monopolies in graphs which are closely related to different parameters in graphs. Given a graph $G=(V,E)$ and $X\subseteq V$, if $\delta_X(v)$ is the number of neighbors $v$ has in $X$, $k$ is an integer and $t$ is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set $M\subseteq V$ a vertex $v$ of $G$ is said to be $k$-controlled by $M$ if $\delta_M(v)\ge \frac{\delta_V(v)}{2}+k$. The set $M$ is called an open $k$-monopoly for $G$ if it $k$-controls every vertex $v$ of $G$. - A function $f: V\rightarrow \{-1,1\}$ is called a signed total $t$-dominating function for $G$ if $f(N(v))=\sum_{v\in N(v)}f(v)\geq t$ for all $v\in V$. - A nonempty set $S\subseteq V$ is a global (defensive and offensive) $k$-alliance in $G$ if $\delta_S(v)\ge \delta_{V-S}(v)+k$ holds for every $v\in V$. In this article we prove that the problem of computing the minimum cardinality of an open $0$-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open $k$-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016

Global defensive k-alliances in graphs

Let $\Gamma=(V,E)$ be a simple graph. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X$. A nonempty set $S\subseteq V$ is a \emph{defensive $k$-alliance} in $\Gamma=(V,E)$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+k,$ $\forall v\in S.$ A defensive $k$-alliance $S$ is called \emph{global} if it forms a dominating set. The \emph{global defensive $k$-alliance number} of $\Gamma$, denoted by $\gamma_{k}^{a}(\Gamma)$, is the minimum cardinality of a defensive $k$-alliance in $\Gamma$. We study the mathematical properties of $\gamma_{k}^{a}(\Gamma)$

Global strategic alliances in scheduled air transport - implications for competition policy

In international aviation, global strategic alliances (GSAs) have in recent years become an important form of cooperation between airlines. This cooperation has hit the antitrust nerve of the European Commission. Initially, the Commission had attempted to constrain both the market share of the major alliances in transatlantic air transport and their access to major European hubs (London and Frankfurt). The airlines maintain that they need alliances as an inevitable means to adapt to the changing environment in increasingly liberalized and globalized air transport markets in order to remain competitive and to fully realize their growth potential. The final verdict by the Commission will be published soon. Though the existing airline alliances are not stable enough to threaten competition and the openness of airline markets on a global scale, certain hubs or even city pairs might be in danger of being dominated by an individual alliance. This is all the more so as alliances in aviation — contrary to, e.g., strategic R&D alliances in manufacturing — are based on cooperating in a core area of the participants' commercial activities, which might end in collusion. On the other hand, alliances may indeed be regarded as an appropriate tool for internationally active firms to remain competitive. For analyzing alliances' impact on competition, networks seem to be more appropriate than city pairs. On the networks level, complementary alliances usually improve overall welfare via lower fares in all submarkets, whereas parallel alliances tend to result in higher prices in the former parallel markets and lower in other markets due to network spillover effects. Since GSAs in aviation are both of a complementary and a parallel nature, no clear-cut a priori position for or against alliances can be maintained based on conventional antitrust reasoning. From the new institutional economics perspective, alliances are ambiguous as well, because this perspective highlights the efficiency objectives of the participating carriers as well as the potential for collusion and opportunistic behavior. Empirical evidence on the market shares and pricing behavior of alliances and their members does not as yet reflect an increasing threat to competition by these forms of cooperation. But it should be noted that alliances appear to be gaining greater stability over time and that the number of independent competitors is shrinking. These independent competitors contribute much to the dynamics of the competitive process. If their vital role for competition were to be restricted, GSAs in airtransport might prove to be detrimental in the long run. The European Commission is right to be on the alert about GSAs having potentially detrimental effects on competition. However, the Commission should avoid overreacting in its zeal to keep markets open (contestable). It should be borne in mind that market access on transatlantic as well as on most other international air transport routes is still governed by the administrative provisions of intergovernmental bilateral agreements and not by market forces. Therefore, the rrtore relevant question for aviation.policy would be whether competition on the North Atlantic routes could be best maintained by scrapping the bilateral agreements and embarking on a truly liberal open skies aviation agreement between the EU, the United States, and other countries. --

A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Sources and Expressions of power in global food coordination and rural sites: domination, counter domination and alternatives

Editorial Introductio

Quest for Aviation Growth

Based on a review of the Airlines Alliances in various parts of the world, the paper comments on the choices before India. It develops the basic contours of strategy for growth of Indian Aviation and notes the role of the government in this context. It suggests a step-wise approach starting with streamlining the domestic aviation scenario, enlarging the market size by treating air travel as an alternative mode rather than as something exclusive for the corporate sector. It also suggests need for adding capacity, creating openness and cooperation among the domestic players in dealing with the international competition. It further recommends creation of maintenance and transit hubs to carve a share of the international aviation business. Finally, it recommends a gradual step-wise offer of equity partnership to preferred partners in the aeroplane and aviation equipment manufacturing industry.