Global defensive alliances in double-loop networks
羅勝鴻, Sheng-Hung Lo · 2008
A defensive alliance in graph G = (V,E) is a set of vertices S in V satisfying |N[v]∩S| ≧ |N(v) ∩ (V - S)| for any v in S, N(v) = {u : uv in E}, and N[v] =N(v)∪{v}. Because of such an alliance, the vertices in S, agreeing to mutually support each other, have the strength of numbers to be able to defend themselves from the vertices in V - S. A defensive alliance S is called global if N[S] = V . A double-loop network DL(n; a , b) can be viewed as a directed graph with n vertices 0,1,2,...,(n,1) and 2n directed edges of the form i -> i+a (mod n) and i -> i+b (mod n), referred to as a-links and b-links. In this thesis, any reference to DL(n; a, b) will mean an underlying graph of a directed graph DL(n; a , b). In this thesis, we study global defensive alliance in DL(n; a, b). We deter- mine the value of the global defensive alliance number in DL(n; 1, 2), DL(n; 1, 3), DL(3n; 1, 3k), and DL(n; 1, n/2). Finally, we research into the relation between γa(G) and integer programming for G being a k-regular graph.