Independent [1,2]-sets in some classes of cactus graphs

• Main Author: Chien, Hans Steven Co
• Format: text
• Language: English
• Published: Animo Repository 2020
• Subjects: Charts, diagrams, etc; Graphic methods; Mathematics
• Online Access: https://animorepository.dlsu.edu.ph/etd_masteral/5984; https://animorepository.dlsu.edu.ph/context/etd_masteral/article/13088/viewcontent/Co_Chien_Hans_Steven_11698306_Partial.pdf
• Institution: De La Salle University

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.