  For each squarefree monomial ideal $I\subset S = k[x_{1},\ldots, x_{n}] $, we
associate a simple graph $G_I$ by using the first linear syzygies of $I$. In
cases, where $G_I$ is a cycle or a tree, we show the following are equivalent:
  (a) $ I $ has a linear resolution
  (b) $ I $ has linear quotients
  (c) $ I $ is a variable-decomposable ideal
  In addition, with the same assumption on $G_I$, we characterize all monomial
ideals with a linear resolution. Using our results, we characterize all
Cohen-Macaulay codimension $2$ monomial ideals with a linear resolution. As an
other application of our results, we also characterize all Cohen-Macaulay
simplicail complexes in cases that $G_{\Delta}\cong G_{I_{\Delta^{\vee}}}$ is a
cycle or a tree.
