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Client–Server and Cost Effective Sets in GraphsChellali, Mustapha, Haynes, Teresa W., Hedetniemi, Stephen T. 01 August 2018 (has links)
For any integer k≥0, a set of vertices S of a graph G=(V,E) is kcosteffective if for every v∈S,N(v)∩(V∖S)≥N(v)∩S+k. In this paper we study the minimum cardinality of a maximal kcosteffective set and the maximum cardinality of a kcosteffective set. We obtain Gallaitype results involving the kcosteffective and global koffensive alliance parameters, and we provide bounds on the maximum kcosteffective number. Finally, we consider kcosteffective sets that are also dominating. We show that computing the kcosteffective domination number is NPcomplete for bipartite graphs. Moreover, we note that not all trees have a kcosteffective dominating set and give a constructive characterization of those that do.

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Partitioning A Graph In Alliances And Its Application To Data ClusteringHassanShafique, Khurram 01 January 2004 (has links)
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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