Ideally Connected Cographs and Chordal Graphs

For distinct vertices $u,v$ in a graph $G$, let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then, $\kappa_G(u,v) \leq \min\{ \mbox{deg}_G(u), \mbox{deg}_G(v) \}$. If equality is attained for every pair of vertices in $G$, then $G$ is called ideally connected. In this paper, we characterize the ideally connected graphs in two well-known graph classes: the cographs and the chordal graphs. We show that the ideally connected cographs are precisely the $2K_2$-free cographs, and the ideally connected chordal graphs are precisely the threshold graphs, the graphs that can be constructed from the single-vertex graph by repeatedly adding either an isolated vertex or a dominating vertex.