Total edge-vertex domination

An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A vertex with degree one is named a leaf. A vertex adjacent to a leaf is named support. The support vertices are classified as weak support if they have degree two or strong support otherwise. If an edge is incident to a leaf, it is named an end edge. If an edge is adjacent to an end edge, it is named a support edge. If diameter of a tree is even, a vertex that is in the middle of the tree is named a central vertex. The diameter of a tree T is denoted by diam(T). An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A vertex v ve-dominates an edge e which is incident to v, as well as every edge adjacent to e.