Xinhong Zhang, Ruiyang Wang, Yaonan Li, Ruijuan Li
Let $D$ be an oriented graph and $X\subseteq V(D)$. The new oriented graph is obtained by reversing all the arcs of $D$ with end-vertices in $X$, then this oriented graph is called the inversion of $X$ in $D$. The inversion number of $D$, denoted as inv(D), is the minimum positive integer $k$ such that $D$ can be transformed into an acyclic oriented graph by successively reversing the vertex subsets $X_1, \dots, X_k$. Let $G$ be an undirected graph and $Y\subseteq V(G)$. If a new graph is obtained by replacing $G[Y]$ with $\overline{G[Y]}$ in $G$, then this graph is called the subgraph complementation of $G$ with respect to $Y$. The minimum positive integer $l$ such that $G$ can be transformed into an empty graph by successively performing the subgraph complementation operation on the vertex subsets $Y_1, \dots, Y_l$ in $G$ is called the subgraph complementation number of $G$, denoted as $c_2(G)$. In this paper, we establish the connection between the inversion number and the subgraph complementation number, and show an upper bound on the inversion number of the $(k+s)$-joins of $k$ tournaments and $s$ oriented graphs under certain conditions, as well as a lower bound on the $k$-joins of oriented graphs. This generalizes the work done by Behague et al. in solving two questions proposed by Bang-Jensen et al. and Aubian et al.
