  The resilience of a Boolean query is the minimum number of tuples that need
to be deleted from the input tables in order to make the query false. A
solution to this problem immediately translates into a solution for the more
widely known problem of deletion propagation with source-side effects. In this
paper, we give several novel results on the hardness of the resilience problem
for $\textit{binary conjunctive queries with self-joins}$ (i.e. conjunctive
queries with relations of maximal arity 2) with one repeated relation. Unlike
in the self-join free case, the concept of triad is not enough to fully
characterize the complexity of resilience. We identify new structural
properties, namely chains, confluences and permutations, which lead to various
$NP$-hardness results. We also give novel involved reductions to network flow
to show certain cases are in $P$. Overall, we give a dichotomy result for the
restricted setting when one relation is repeated at most 2 times, and we cover
many of the cases for 3. Although restricted, our results provide important
insights into the problem of self-joins that we hope can help solve the general
case of all conjunctive queries with self-joins in the future.
