  The Connected Vertex Cover problem is to decide if a graph G has a vertex
cover of size at most $k$ that induces a connected subgraph of $G$. This is a
well-studied problem, known to be NP-complete for restricted graph classes,
and, in particular, for $H$-free graphs if $H$ is not a linear forest (a graph
is $H$-free if it does not contain $H$ as an induced subgraph). It is easy to
see that Connected Vertex Cover is polynomial-time solvable for $P_4$-free
graphs. We continue the search for tractable graph classes: we prove that it is
also polynomial-time solvable for $(sP_1+P_5)$-free graphs for every integer
$s\geq 0$.
