  The type-I intermittency route to (or out of) chaos is investigated within
the Horizontal Visibility graph theory. For that purpose, we address the
trajectories generated by unimodal maps close to an inverse tangent bifurcation
and construct, according to the Horizontal Visibility algorithm, their
associated graphs. We show how the alternation of laminar episodes and chaotic
bursts has a fingerprint in the resulting graph structure. Accordingly, we
derive a phenomenological theory that predicts quantitative values of several
network parameters. In particular, we predict that the characteristic power law
scaling of the mean length of laminar trend sizes is fully inherited in the
variance of the graph degree distribution, in good agreement with the numerics.
We also report numerical evidence on how the characteristic power-law scaling
of the Lyapunov exponent as a function of the distance to the tangent
bifurcation is inherited in the graph by an analogous scaling of the block
entropy over the degree distribution. Furthermore, we are able to recast the
full set of HV graphs generated by intermittent dynamics into a renormalization
group framework, where the fixed points of its graph-theoretical RG flow
account for the different types of dynamics. We also establish that the
nontrivial fixed point of this flow coincides with the tangency condition and
that the corresponding invariant graph exhibit extremal entropic properties.
