Erratum to "Total domination supercritical graphs with respect to relative complements" [Discrete Math. 258 (2002) 361-371]

A total dominating set of a graph G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total domination number of G, denoted by γt(G), is the minimum cardinality of a total dominating set. Let G be a connected spanning subgraph of Ks,s and letH be the complement of G relative to Ks,s; that is, Ks,s = G⊕H . The graph G is k-supercritical relative to Ks,s if γt(G) = k and γt(G + e) = k − 2 for all e ∈ E(H). In Section 4 in [1], it is shown that the diameter of a 6-supercritical graph is 3, 4 or 5. Further 6-supercritical graphs with diameter 4 or 5 are characterized. As a concluding remark a construction of 6-supercritical graphs with diameter 3 is described. However this construction is incorrect. Here we provide a correct construction. Following the notation in [1], if all edges between two independent sets X and Y are present, we say that [X, Y ] is full. For k ≥ 3, let Gk be the graph with vertex set A ∪ B ∪ C ∪ D ∪ {x, y}, where A = {a1, a2, . . . , ak}, B = {b1, b2, . . . , bk}, C = {c1, c2, . . . , ck} and D = {d1, d2, . . . , dk}, and with edge set E(Gk) = {xai | i = 1, 2, . . . , k} ∪ {ydi | i = 1, 2, . . . , k}∪{aibj | 1 ≤ i, j ≤ k, i ≠ j}∪{bicj | 1 ≤ i, j ≤ k, i ≠ j}∪{cidj | 1 ≤ i, j ≤ k, i ≠ j}∪{aidi | i = 1, 2, . . . , k}∪{xy}. Thus, [{x}, A] and [{y},D] are full, each of [A, B], [B, C] and [C,D] is full minus a perfect matching, while [A,D] consists of a perfect matching. The graph Gk is a 6-supercritical graph with diameter 3.