A linear algorithm for secure domination in trees

The Secure Domination Number of Cartesian Products of Small Graphs with Paths and Cycles

“…Algorithms exist to solve the secure domination problem, including linear-time algorithms for trees [5] and block graphs [17], as well as algorithms for general graphs [2,3,4]. However, there are very few infinite families of graphs for which the secure domination number is known.…”

“…In the former case, v 2 2 is left unguarded. In the latter case, v 5 1 is left unguarded. This implies that S is not secure dominating, which is a contradiction, and hence S cannot contain the pattern 101101. .…”

“…Then, if there is an attack at v 3 2 , the guard at v 4 2 must move to assist, which implies that v 6 2 ∈ S in order to avoid leaving v 5 2 unguarded. Likewise, if there is an attack at v 3 1 , the guard at v 2 1 cannot move to assist without leaving v 2 2 unguarded.…”

“…In the former case, v 2 1 is left unguarded. In the latter case, v 5 2 is left unguarded. Hence, this is impossible, and S cannot contain the pattern 0110.…”

“…Then, if there is an attack at v 5 2 , the guard at v 6 2 must move to assist, which implies that v 8 2 ∈ S in order to avoid leaving v 7 2 unguarded. Then, if there is an attack at v 7 2 , the guard at v 6 2 cannot move to assist without leaving v 5 2 unguarded. Therefore, the guard at v 8 2 must move to assist, which implies that v 9 1 ∈ S in order to avoid leaving v 8 1 unguarded.…”