Eternal Security in Graphs

Consider placing a guard on each vertex of a dominating set \( S_0 \) of a graph. If for every vertex \( v \notin S_0 \), there is a corresponding guard at an adjacent vertex \( u \) for which the resulting set \( S_1 = S_0 – \{u\} \cup \{v\} \) is dominating, then we say that \( S_0 \) is \( 1 \)-secure. It is eternally \( 1 \)-secure if for any sequence \( v_1, v_2, \ldots, v_k \) of vertices, there exists a sequence of guards \( u_1, u_2, \ldots, u_k \) with \( u_i \in S_{i-1} \) and \( u_i \) equal to or adjacent to \( v_i \), such that each set \( S_i = S_{i-1} – \{u_i\} \cup \{v_i\} \) is dominating. We investigate the minimum cardinality of an eternally secure set. In particular, we refute a conjecture of Burger et al. We also investigate eternal \( m \)-security, in which all guards can move simultaneously.