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We consider the problem of computing the size of each $r$-neighbourhood forevery vertex of a graph. Specifically, we ask whether the size of the closedsecond neighbourhood can be computed in subquadratic time.  Adapting the SETH reductions by Abboud et al. (2016) that excludesubquadratic algorithms to compute the radius of a graph, we find that asubquadratic algorithm would violate the SETH. On the other hand, a linearfpt-time algorithm by Demaine et al. (2014) parameterized by a certain`sparseness parameter' of the graph is known, where the dependence on theparameter is exponential. We show here that a better dependence is unlikely:for any~$\delta < 2$, no algorithm running in time $O(2^{o({\rm vc}(G))} \,n^{\delta})$, where~${\rm vc}(G)$ is the vertex cover number, is possibleunless the SETH fails.  We supplement these lower bounds with algorithms that solve the problem intime~$O(2^{{\rm vc}(G)/2} {\rm vc}(G)^2 \cdot n)$ and $O(2^w w \cdot n)$.

Lower and Upper Bound for Computing the Size of All Second Neighbourhoods