2019/09/22 by Anderson, Sarah E., Kuenzel, Kirsti, Rall, Douglas F. · 1 citation
#05C69 #05C76 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1909.09955
A graph is well-dominated if all of its minimal dominating sets have the same cardinality. We prove that at least one of the factors is well-dominated if the Cartesian product of two graphs is well-dominated. In addition, we show that the Cartesian product of two connected, triangle-free graphs is well-dominated if and only if both graphs are complete graphs of order 2. Under the assumption that at least one of the connected graphs G or H has no isolatable vertices, we prove that the direct product of G and H is well-dominated if and only if either G=H=K3 or G=K2 and H is either the 4-cycle or the corona of a connected graph. Furthermore, we show that the disjunctive product of two connected graphs is well-dominated if and only if one of the factors is a complete graph and the other factor has domination number at most 2.