Secure and secure-dominating sets of graphs

• Main Authors: Yen-Hsuan Chang, 張晏瑄
• Other Authors: David Kuo
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/07842213271087974505

碩士 === 國立東華大學 === 應用數學系 === 97 === Given a graph G and a set S ⊆ V (G), a function A : S → P(V (G) − S) is called an attack on S (in G) if A(u) ⊆ N (u)−S for all u ∈ S and A(u)∩A(v) = ∅ for all u, v ∈ S, u 6= v. And a function D : S → P(S) is called a defense of S if D(u) ⊆ N [u] ∩ S for all u ∈ S and D(u) ∩ D(v) = ∅ for all u, v ∈ S, u 6= v. For a set S and an attack A on S, a defense of S is called a defense of S corresponding to A (in G) if |D(u)| ≥ |A(u)| for all u ∈ S. A nonempty set S ⊆ V (G) is called a secure set of G if for all attack A on S, there exists a defense of S corresponding to A. The cardinality of a minimum secure set in graph G is the security number of G and is denoted s(G). For a given graph G, a set S ⊆ V (G) is called a dominating set of G if N [S] = V (G). And we say that S is a secure-dominating set of G if S is a secure set of G that is also a dominating set of G. The secure-domination number of G, denoted γs(G), is defined by γs(G) = min{|S| : S is a secure-dominating set of G}. In this thesis, we study the secure-domination number and the secure number of graphs. We give a lower bound for the secure-domination number of graphs, and find the secure-domination number of the complete k-partite graphs and the Cartesian product of paths and cycles. And we also find the secure number of those graphs G that are the Cartesian product of a power of a path and a path.