Let's define a graph manifold as 3-manifold which is obtained by plumbing of a circle bundle over $\Sigma _g$ (with Euler number 0) with a circle bundle over $\Sigma_h$ (with Euler number k). Is there any way to find an open book decomposition of this manifold explicitly?(I think this is the same as asking how to switch between a spinal open book decomposition of a 3-manifold and an open book decomposition of it)
One can produce open books for graph manifolds in basically the same way that one produces an open book when cabling a fibered knot. Let's begin there.
With a fibered knot you've got a torus $T$ that separates a $D^2 \times S^1$ neighborhood of the knot from a surface bundle over the circle. Near $T$, these both look like fibrations that each meet $T$ in different slopes. An essential simple closed curve in $T$ with slope different from these two gives the binding for an open book that `interpolates' between $D^2$ fibers and surface fibers. Outside a neighborhood of $T$, the pages of this open book are comprised of $D^2$ fibers and surface fibers. Further details of such a construction are in http://arxiv.org/abs/1005.1978 for example. Here's a picture of this sort of thing: http://sketchesoftopology.wordpress.com/2009/11/18/cabling-a-knots-surface/
This works equally well when $T$ (or a collection of tori) decomposes your manifold into surface bundles over the circle whose fibers meet in different slopes on $T$, like when you have a spinal open book. Do this cabling procedure along all your decomposing tori with a bit of care, and you'll get an open book.
For a graph manifold where $T$ decomposes the manifold into circle bundles (or more generally Seifert fibered spaces) you just need to further decompose along tori that cut off $D^2 \times S^1$ neighborhoods of enough circle fibers until the resulting pieces are all surface bundles. In this case you'll end up with the pieces all being a surface times a circle. Do this sort of cabling procedure again.
(And I reckon if the base space of one of your Seifert fibered spaces is non-orientable, you'll have to work a wee bit harder. But it's doable.)