  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. Given $s,t\in V(G)$, an
$st$-bond of $G$ is a bond whose removal disconnects $s$ and $t$. Contrasting
with the large number of studies related to maximum cuts, there are 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 largest
$st$-bond of a graph. Although cuts and bonds are similar, we remark that
computing the largest bond of a graph tends to be harder than computing its
maximum cut. We show that {\sc Largest Bond} remains NP-hard even for planar
bipartite graphs, and it does not admit a constant-factor approximation
algorithm, unless $P = NP$. We also show that {\sc Largest Bond} and {\sc
Largest $st$-Bond} on graphs of clique-width $w$ cannot be solved in time
$f(w)\times n^{o(w)}$ unless the Exponential Time Hypothesis fails, but they
can be solved in time $f(w)\times n^{O(w)}$. In addition, we show that both
problems are fixed-parameter tractable when parameterized by the size of the
solution, but they do not admit polynomial kernels unless NP $\subseteq$
coNP/poly.
