Power domination of the cartesian product of graphs

On the power domination number of the Cartesian product of graphs

“…Finally to show equality, we observe that when m ≥ n, the set {(i, 0) | 1 ≤ i ≤ n − 1} with cardinality n − 1 is a PDS of K n □ K 1,m , and when n ≥ m, the set {(1, j) | 1 ≤ j ≤ m − 1} with cardinality m − 1 is a PDS of K n □ K 1,m . □ With Theorems 3, 6, 9, 10 and the results presented in [8] and [9], we summarize the power domination numbers involving the Cartesian product in Table 1.…”

Power Domination in Graphs

“…It should be noted that a recent survey of results on graph products is given by Soh and Koh in [27], which is more detailed than what we present here, and should be referred to for having an exhaustive list of theorems on the topic. The same authors also surveyed earlier the results on the Cartesian product in [22].…”

The fixed set search applied to the power dominating set problem

“…It has been shown that the PDSP is NP‐hard even on specific types of graphs like cubic graphs (Binkele‐Raible & Fernau, 2012). Another direction of research has been in finding the optimal size or bounds for specific graphs like grid graphs (Dorfling & Henning, 2006; Pai, Chang, & Wang, 2007), hyper cubes (Dean, Ilic, Ramirez, Shen, & Tian, 2011), generalised Petersen graphs (Koh & Soh, 2016; Lai, Chien, Chou, & Kao, 2012), cylinders (Koh & Soh, 2016), circular‐arc graphs (Liao & Lee, 2013), and tori (Koh & Soh, 2016). The PDSP has been generalised in the form of the k ‐power dominating set problem ( k ‐PDSP) which analyses the potential extensions of the propagation rule in the original problem (Chang, Dorbec, Montassier, & Raspaud, 2012).…”

“…, cylinders (Koh & Soh, 2016), circular-arc graphs (Liao & Lee, 2013), and tori (Koh & Soh, 2016). The PDSP has been generalised in the form of the k-power dominating set problem (k-PDSP) which analyses the potential extensions of the propagation rule in the original problem (Chang, Dorbec, Montassier, & Raspaud, 2012).…”