  In this work we introduce the graph-theoretic notion of mendability: for each
locally checkable graph problem we can define its mending radius, which
captures the idea of how far one needs to modify a partial solution in order to
"patch a hole."
  We explore how mendability is connected to the existence of efficient
algorithms, especially in distributed, parallel, and fault-tolerant settings.
It is easy to see that $O(1)$-mendable problems are also solvable in $O(\log^*
n)$ rounds in the LOCAL model of distributed computing. One of the surprises is
that in paths and cycles, a converse also holds in the following sense: if a
problem $\Pi$ can be solved in $O(\log^* n)$, there is always a restriction
$\Pi' \subseteq \Pi$ that is still efficiently solvable but that is also
$O(1)$-mendable.
  We also explore the structure of the landscape of mendability. For example,
we show that in trees, the mending radius of any locally checkable problem is
$O(1)$, $\Theta(\log n)$, or $\Theta(n)$, while in general graphs the structure
is much more diverse.
