Disjunctive total domination in permutation graphs

Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G). If G has no isolated vertex, then a disjunctive total dominating set (DTD-set) of G is a vertex set S ⊆ V (G) such that every vertex in G is adjacent to a vertex of S or has at least two vertices in S at distance two from it, and the disjunctive total domination number γtd(G) of G is the minimum cardinality overall DTD-sets of G. Let G1 and G2 be two disjoint copies of a graph G, and let σ : V (G1) → V (G2) be a bijection. Then, a permutation graph Gσ = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪{uv|v = σ(u)}. For any connected graph G of order at least three, we prove the sharp bounds 2 ≤ γtd(G σ) ≤ 2γtd(G); we give an example showing that γtd(G) − γ td(G σ) can be arbitrarily large. We characterize permutation graphs for which γtd(G σ) = 2 holds. Further, we show that γtd(G σ) ≤ 2γtd(G) − 1 when G is a cycle, a path, and a complete k-partite graph, respectively.