Domination and edge-vertex domination in trees

Let [Formula: see text] be a simple connected graph. An edge [Formula: see text] [Formula: see text]-dominates a vertex [Formula: see text] if [Formula: see text] is incident with [Formula: see text] or [Formula: see text] is incident with a vertex adjacent to [Formula: see text]. A set [Formula: see text] is an edge-vertex dominating set of graph [Formula: see text] if every vertex in [Formula: see text] is [Formula: see text]-dominated by an edge in [Formula: see text]. An edge-vertex dominating set [Formula: see text] is minimal edge-vertex dominating set if any proper subset of [Formula: see text] is not an edge-vertex dominating set. The minimum (maximum, respectively) cardinality of minimal edge-vertex dominating set is called the edge-vertex domination number (upper edge-vertex domination number, respectively) of a graph [Formula: see text] and is denoted by [Formula: see text] ([Formula: see text], respectively). Gallai type inequalities [Formula: see text] and [Formula: see text] are proved and extremal trees attaining the equality are characterized.