ABSTRACT
Graph theory plays a vital role in medical fields, especially the theory of domination in graphs. The concept of eternal domination in trees was introduced by W. F. Klostermeyer et. al. A subset D of a vertex set in G is said to be an eternal dominating set, if for all possible sequences of attacks R = r 1, r 2, r 3, ..., there exists a sequence D = D 1, D 2, D 3, ..., of dominating sets such that D i+1 = (D o \{v}) ∪ {ri }, where v ∊ Di and ri ∊ N[v]. The set Di is the set of locations of the guards after attacks at ri are defended. If v ≠ ri , we say that the guard at v has moved to ri . The minimum cardinality of these eternal dominating sets in G is called the eternal domination number in G and is denoted by γ∞ (G). This concept plays a vital role to cure diseases in a particular area and control them. In this chapter, it is proposed to provide a real life application in epidemiology. Also, we obtain this number for various product related graphs.
