Middle domination and 2-independence in trees

Abstract

Let [Formula: see text] be a finite simple graph. The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. The middle domination number [Formula: see text] of [Formula: see text] is defined by the domination number [Formula: see text] of the middle graph [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These parameters are incomparable in general. However, we show that [Formula: see text] for any tree [Formula: see text]. We also characterize all trees attaining the equality.