  A matching $M$ in a graph $\Gamma$ is positive if $\Gamma$ has a
vertex-labeling such that $M$ coincides with the set of edges with positive
weights. A positive matching decomposition (pmd) of $\Gamma$ is an
edge-partition $M_1,\ldots,M_p$ of $\Gamma$ such that $M_i$ is a positive
matching in $\Gamma-M_1\cup\cdots\cup M_{i-1}$, for $i=1,\ldots,p$. The pmds of
graphs are used to study algebraic properties of the Lov\'{a}sz-Saks-Schrijver
ideals arising from orthogonal representations of graphs. We give a
characterization of pmds of graphs in terms of alternating closed walks and
apply it to study pmds of various classes of graphs including complete
multipartite graphs, (regular) bipartite graphs, cacti, generalized Petersen
graphs, etc. We further show that computation of pmds of a graph can be reduced
to that of its maximum pendant-free subgraph.
