  A graph $H$ is said to be {\em light} in a family $\mathfrak{G}$ of graphs if
at least one member of $\mathfrak{G}$ contains a copy of $H$ and there exists
an integer $\lambda(H, \mathfrak{G})$ such that each member $G$ of
$\mathfrak{G}$ with a copy of $H$ also has a copy $K$ of $H$ such that
$\deg_{G}(v) \leq \lambda(H, \mathfrak{G})$ for all $v \in V(K)$. In this
paper, we study the light graphs in the class of graphs with small average
degree, including the plane graphs with some restrictions on girth.
