A dominating set in a graph is a set S of vertices such that every vertex not in S is adjacent to a member S. A global defensive alliance in a graph is a dominating set S with the property that every vertex in S has in its closed neighborhood at least as many vertices in S as in V -- S. A global offensive alliance in a graph is a dominating set S with the property that every vertex in V -- S has in its closed neighborhood at least as many vertices in S as in V -- S. / We first study alliances in graphs. There are many results in this area. For example, it is known that every connected n-vertex graph G has a global offensive alliance of size at most 2n/3. Another result is that any global defensive alliance in G has size at least ( 4n+1 - 1)/2. Our study focuses particularly on trees and algorithms for finding alliances in trees. We find the cardinality of a minimum global offensive alliance for complete k-ary trees and the minimum cardinality global defensive alliance for complete binary and ternary trees. Also, we present a linear time algorithm for finding the minimum global offensive alliance in a tree. Additionally, for a general graph, an upper bound is given on the size of a minimum global offensive alliance in terms its degree sequence. The methods that we use in this part of the thesis are mainly algorithmic and deterministic. / Then we study independent dominating sets in graphs. An independent dominating set is a set of mutually nonadjacent vertices which is also a dominating set. This is a well-studied topic (see Haynes, Hedetniemi and Slater ). Our main result is to show that every d-regular graph of order n with girth at least 5 and satisfying d = o(1) and d2(log d)3/2 = o(n) contains an independent dominating set of size at most (1 + o(1)) nlogdd . This generalizes the results of Duckworth and Wormald  for random regular graphs. We construct the independent dominating set using recent probabilistic methods which resemble Rodl's semi-random method (see for example Alon and Spencer ).
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