On the geodetic number of permutation graphs

A vertex set S in a graph G is a geodetic set if every vertex of G lies on some u?v geodesic of G, where u,v∈S. The geodetic number g(G) of G is the minimum cardinality over all geodetic sets of G. Let G ₁ and G ₂ be disjoint copies of a graph G, and let σ:V(G ₁)→V(G ₂) be a bijection. Then, a permutation graph G _σ =(V,E) has the vertex set V=V(G ₁)∪V(G ₂) and the edge set E=E(G ₁)∪E(G ₂)∪{uv∣v=σ(u)}. For any connected graph G of order n≥3, we prove the sharp bounds 2≤g(G _σ )≤2n?(1+△(G)), where △(G) denotes the maximum degree of G. We give examples showing that neither is there a function h ₁ such that g(G)<h ₁(g(G _σ )) for all pairs (G,σ), nor is there a function h ₂ such that h ₂(g(G))>g(G _σ ) for all pairs (G,σ). Further, we characterize permutation graphs G _σ satisfying g(G _σ )=2|V(G)|?(1+△(G)) when G is a cycle, a tree, or a complete k-partite graph.