The behavior of users in social networks is often observed to be affected by
the actions of their friends. Bhawalkar et al. \cite{bhawalkar-icalp}
introduced a formal mathematical model for user engagement in social networks
where each individual derives a benefit proportional to the number of its
friends which are engaged. Given a threshold degree $k$ the equilibrium for
this model is a maximal subgraph whose minimum degree is $\geq k$. However the
dropping out of individuals with degrees less than $k$ might lead to a
cascading effect of iterated withdrawals such that the size of equilibrium
subgraph becomes very small. To overcome this some special vertices called
"anchors" are introduced: these vertices need not have large degree. Bhawalkar
et al. \cite{bhawalkar-icalp} considered the \textsc{Anchored $k$-Core}
problem: Given a graph $G$ and integers $b, k$ and $p$ do there exist a set of
vertices $B\subseteq H\subseteq V(G)$ such that $|B|\leq b, |H|\geq p$ and
every vertex $v\in H\setminus B$ has degree at least $k$ is the induced
subgraph $G[H]$. They showed that the problem is NP-hard for $k\geq 2$ and gave
some inapproximability and fixed-parameter intractability results. In this
paper we give improved hardness results for this problem. In particular we show
that the \textsc{Anchored $k$-Core} problem is W[1]-hard parameterized by $p$,
even for $k=3$. This improves the result of Bhawalkar et al.
\cite{bhawalkar-icalp} (who show W[2]-hardness parameterized by $b$) as our
parameter is always bigger since $p\geq b$. Then we answer a question of
Bhawalkar et al. \cite{bhawalkar-icalp} by showing that the \textsc{Anchored
$k$-Core} problem remains NP-hard on planar graphs for all $k\geq 3$, even if
the maximum degree of the graph is $k+2$. Finally we show that the problem is
FPT on planar graphs parameterized by $b$ for all $k\geq 7$.Comment: To appear in AAAI 201