Independent [1,2]-sets in some classes of cactus graphs - Chien, Hans Steven Co (Animo Repository, 2020)
A [1, 2]-set S in a graph G is a vertex subset such that every vertex not in S has at least one and at most two neighbours in it. If the additional requirement that the set be independent is added, the existence of such a set is not guaranteed in every graph. In this paper, we study the existence of independent [1, 2]-sets in some classes of cactus graphs and determine such sets for some parameters of the graph. In particular, we will show that there exists an independent [1, 2]-set for any cactus graph with k ≥ 2 cycles, 2 and 3 as the minimum and maximum degree of a vertex in the cactus graph, respectively. We also study the minimum cardinality of an independent [1, 2]-set in some other classes of cactus graphs.