Construction for Trees without vertices contained in all minimum dominating sets
WS Zhao, Xuelu Gao, Heping Zhang · Lanzhou University Institutional Repository · 2018
A vertex v of a graph G is called universal if v belongs to every minimum dominating set of G, idle if v does not belong to any minimum dominating set of G, and alterable if v is neither universal nor idle. A question is: what are the constructions of trees exactly containing one or two kinds of these vertices? This paper intends to improve and perfect the study on this question. We give the constructions for trees only containing non-universal vertices and exactly containing both alterable and idle vertices, respectively. We also point out that trees exactly containing both universal and alterable vertices do not exist.