  We present a detailed study of a specific class of graph that can potentially
contribute to the proton-proton double parton scattering (DPS) cross section.
These are the `2v1' or `single perturbative splitting' graphs, in which two
`nonperturbatively generated' (NP) ladders interact with two ladders that have
been generated via a perturbative 1->2 branching process. Using a detailed
calculation, we confirm the result first written down by Ryskin and Snigirev,
that 2v1 graphs in which the two NP ladders do not interact with one another do
contribute to the LO proton-proton DPS cross section, albeit with a different
geometrical prefactor to the one that applies to the `2v2'/`zero perturbative
splitting' graphs. We then show that 2v1 graphs in which the NP ladders
exchange partons with one another also contribute to the LO proton-proton DPS
cross section, provided that this `crosstalk' occurs at a lower scale than the
1->2 branching on the other side of the graph. Due to the preference in the 2v1
graphs for the x value at which the branching occurs/crosstalk ceases to be
much larger than x values at the hard scale, the effect of crosstalk
interactions is likely to be a decrease in the 2v1 cross section except at very
small x values (< 10^-6). At moderate x values ~10^-3 - 10^-2, the x value at
the splitting is in the region ~10^-1 where PDFs do not change much with scale,
and crosstalk effects are likely to be small. We give an explicit formula for
the contribution from the 2v1 graphs to the DPS cross section, and combine this
with the suggestion that `double perturbative splitting'/`1v1' graphs should be
removed from this cross section made in a previous publication, to obtain a
formula for the DPS cross section. We point out that there are two potentially
concerning features in this equation, that might indicate that our prescription
for handling the 1v1 graphs is not quite correct.
