2,842 research outputs found

    A Survey on Alliances and Related Parameters in Graphs

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    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, Ξ±\alpha-domination, Ξ±\alpha-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global rr-alliances in graphs.We also propose a new framework, called (global) (D,O)(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

    Open k-monopolies in graphs: complexity and related concepts

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    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 kk-monopolies in graphs which are closely related to different parameters in graphs. Given a graph G=(V,E)G=(V,E) and XβŠ†VX\subseteq V, if Ξ΄X(v)\delta_X(v) is the number of neighbors vv has in XX, kk is an integer and tt is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set MβŠ†VM\subseteq V a vertex vv of GG is said to be kk-controlled by MM if Ξ΄M(v)β‰₯Ξ΄V(v)2+k\delta_M(v)\ge \frac{\delta_V(v)}{2}+k. The set MM is called an open kk-monopoly for GG if it kk-controls every vertex vv of GG. - A function f:Vβ†’{βˆ’1,1}f: V\rightarrow \{-1,1\} is called a signed total tt-dominating function for GG if f(N(v))=βˆ‘v∈N(v)f(v)β‰₯tf(N(v))=\sum_{v\in N(v)}f(v)\geq t for all v∈Vv\in V. - A nonempty set SβŠ†VS\subseteq V is a global (defensive and offensive) kk-alliance in GG if Ξ΄S(v)β‰₯Ξ΄Vβˆ’S(v)+k\delta_S(v)\ge \delta_{V-S}(v)+k holds for every v∈Vv\in V. In this article we prove that the problem of computing the minimum cardinality of an open 00-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 kk-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016

    The Power of Small Coalitions under Two-Tier Majority on Regular Graphs

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    In this paper, we study the following problem. Consider a setting where a proposal is offered to the vertices of a given network GG, and the vertices must conduct a vote and decide whether to accept the proposal or reject it. Each vertex vv has its own valuation of the proposal; we say that vv is ``happy'' if its valuation is positive (i.e., it expects to gain from adopting the proposal) and ``sad'' if its valuation is negative. However, vertices do not base their vote merely on their own valuation. Rather, a vertex vv is a \emph{proponent} of the proposal if the majority of its neighbors are happy with it and an \emph{opponent} in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever the majority of its vertices are proponents. We study this problem for regular graphs with loops. Specifically, we consider the class Gn∣d∣h\mathcal{G}_{n|d|h} of dd-regular graphs of odd order nn with all nn loops and hh happy vertices. We are interested in establishing necessary and sufficient conditions for the class Gn∣d∣h\mathcal{G}_{n|d|h} to contain a labeled graph accepting the proposal, as well as conditions to contain a graph rejecting the proposal. We also discuss connections to the existing literature, including that on majority domination, and investigate the properties of the obtained conditions.Comment: 28 pages, 8 figures, accepted for publication in Discrete Applied Mathematic

    Graph Algorithms and Applications

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    The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity

    Partitioning A Graph In Alliances And Its Application To Data Clustering

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    Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets
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