We provide a simple constructive characterization for trees with equal domination and independent domination numbers, and for trees with equal domination and total domination numbers. We also consider a general framework for constructive characterizations for other equality problems.

In Section 5 of [1], class H2 should be defined as follows: Let H2 be the family of graphs whose vertex set contains a set D that induces a star K1,r (with r ≥ 1), of center vertex x and leaves x1, . . . , xr, such that: for every vertex u of V (G) − D, N(u) is equal to either {x1, x2} or {x, xi} for some i ∈ {1, . . . , r}; moreover, if any vertex z of V (G)−D satisfies N(z) = {x1, x2}, then n...

We consider finite graphs G with vertex set V (G). A subset D ⊆ V (G) is a dominating set of the graph G, if every vertex v ∈ V (G) − D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality among the dominating sets of G. In this note, we characterize the trees T with an even number of vertices such that γ(T ) = |V (T )| − 2

Denote the total domination number of a graph G by γt(G). A graph G is said to be total domination edge critical, or simply γtcritical, if γt(G + e) < γt(G) for each edge e ∈ E(G). For 3t-critical graphs G, that is, γt-critical graphs with γt(G) = 3, the diameter of G is either 2 or 3. We characterise the 3t-critical graphs G with diam G = 3.