Movable resolving domination in graphs
Gerald Bacon Monsanto, Helen Moso Rara · Discrete Mathematics Algorithms and Applications · 2021
Let G be a connected graph. Brigham et al., Resolving domination in graphs, Math. Bohem. 1 (2003) 25–36 defined a resolving dominating set as a set S of vertices of a connected graph G that is both resolving and dominating. A resolving dominating is a 1-movable resolving dominating set of G if for every [Formula: see text], either [Formula: see text] is a resolving dominating set or there exists a vertex [Formula: see text] such that [Formula: see text] is a resolving dominating set of G. The minimum cardinality of a 1-movable resolving dominating set of G, denoted by [Formula: see text] is the 1-movable R-domination number of G. A 1-movable resolving dominating set with cardinality [Formula: see text] is called a [Formula: see text]-set of G. In this paper, we characterize the 1-movable resolving dominating sets in the join and lexicographic product of two graphs and determine the bounds or exact values of the 1-movable resolving domination number of these graphs.