Connected minimum secure-dominating sets in grids

Peer review under responsibility of Kalasalingam University.

Figures & data

Fig. 7 A connected secure-dominating set in both $P_9\square P_{10}$ and $P_9\square C_{10}$, on 45 vertices. Edges are omitted.

Fig. 8 A connected secure-dominating set on 18 vertices in $C_6\square C_6$. Edges are omitted.

Fig. 9 A connected secure-dominating set on 25 vertices in $C_7\square C_7$. Edges are omitted.

Fig. 10 A connected secure-dominating set $C_7\square C_9$. Edges are omitted.

Fig. 1 Case 3. Vertices in $S$ are blacked-in.

Fig. 2 A connected minimum secure dominating set of vertices in $P_8\square P_7$ with $28=\frac{8\times 7}{2}$ vertices.

Fig. 3 $P_6\square P_n$, a minimum secure dominating set which is also connected.

Fig. 4 A connected secure dominating set in $P_9\square P_{11}$.

Fig. 5 A connected secure dominating set in $P_{11}\square P_9$ with $50=\lceil \frac{99}{2}\rceil$ vertices.

Fig. 6 A connected secure-dominating set in both $P_5\square P_6$ and $P_5\square C_6$, on 15 vertices. Edges are omitted.