  The cut-set $\partial(S)$ of a graph $G=(V,E)$ is the set of edges that have
one endpoint in $S\subset V$ and the other endpoint in $V\setminus S$, and
whenever $G[S]$ is connected, the cut $[S,V\setminus S]$ of $G$ is called a
connected cut. A bond of a graph $G$ is an inclusion-wise minimal disconnecting
set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of
$G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Contrasting with
a large number of studies related to maximum cuts, there exist very few results
regarding the largest bond of general graphs. In this paper, we aim to reduce
this gap on the complexity of computing the largest bond, and the maximum
connected cut of a graph. Although cuts and bonds are similar, we remark that
computing the largest bond and the maximum connected cut of a graph tends to be
harder than computing its maximum cut. We show that it does not exist a
constant-factor approximation algorithm to compute the largest bond, unless P =
NP. Also, we show that {\sc Largest Bond} and {\sc Maximum Connected Cut} are
NP-hard even for planar bipartite graphs, whereas \textsc{Maximum Cut} is
trivial on bipartite graphs and polynomial-time solvable on planar graphs. In
addition, we show that {\sc Largest Bond} and {\sc Maximum Connected Cut} are
NP-hard on split graphs, and restricted to graphs of clique-width $w$ they can
not be solved in time $f(w)\times n^{o(w)}$ unless the Exponential Time
Hypothesis fails, but they can be solved in time $f(w)\times n^{O(w)}$.
Finally, we show that both problems are fixed-parameter tractable when
parameterized by the size of the solution, the treewidth, and the twin-cover
number.
