Power domination in generalized Mycielskian of spiders - Sreethu, K; Varghese, Seema (MATHEMATICA SCANDINAVICA, 2024)
The power domination problem in the graph was introduced to model the monitoring problem in electric networks. It is to find a set of vertices with minimum cardinality, called a power dominating set (PDS), that monitors the whole set $ V $ after applying the rules, domination and propagation. In this paper, we discuss the power domination problem in generalized Mycielskian of spiders. We characterize spiders with $γ _P(μ _m(T))= 1$. If $γ _P(μ _m(T)) eq 1 $ and $ m $ is even then $ γ _P(μ _m(T))=\frac {m}{2} +1$. We also show that if $ m $ is odd and $ γ _P(μ _m(T)) eq 1 $, then $ \lceil \frac {m+1}{2}\rceil ≤ γ _P(μ _m(T))≤ \lceil \frac {m}{2}\rceil +1$.