Inversion Diameter and Treewidth

In an oriented graph (G) over right arrow, the inversion of a subset X of vertices is the operation that reverses the orientation of all arcs with both end-vertices in X. The inversion graph of a graph G, denoted by I(G), is the graph whose vertices are orientations of G in which two orientations (G) over right arrow(1) and (G) over right arrow(2) are adjacent if and only if there is an inversion transforming (G) over right arrow(1) into (G) over right arrow(2). The inversion diameter of a graph G is the diameter of its inversion graph I(G), denoted by diam(I(G)). Havet, Horsch, and Rambaud (2024) first proved that for G of treewidth k, diam(Z(G)) <= 2k, and that there are graphs of treewidth k with inversion diameter k + 2 In this paper, we construct graphs of treewidth k with inversion diameter 2k, which implies that the previous upper bound diam(I(G)) <= 2k is tight. Moreover, for graphs with maximum degree Delta, Havet, Horsch, and Rambaud (2024) proved diam(I(G)) <= 2 Delta - 1 and conjectured that diam(I(G)) <= Delta. We prove the conjecture when Delta = 3 with the help of computer calculations.