  Given a graph $G$, the burning number of $G$ is the smallest integer $k$ for
which there are vertices $x_1, x_2,\ldots,x_k$ such that $(x_1,x_2,\ldots,x_k)$
is a burning sequence of $G$. It has been shown that the graph burning problem
is NP-complete, even for trees with maximum degree three, or linear forests. A
$t$-unicyclic graph is a unicycle graph with exactly one vertex of degree
greater than $2$. In this paper, we first present the bounds for the burning
number of $t$-unicyclic graphs, and then use the burning numbers of linear
forests with at most three components to determine the burning number of all
$t$-unicyclic graphs for $t\le 2$.
