This well-cited paper talks about is a minimal renormalizable extension to the Standard Model (SM) to incorporate particle dark matter (DM) into it by adding a real scalar field $S$ which (unlike the Higgs doublet $H$) is a singlet under the full SM gauge group. But we have to pay a price, we have to introduce three more free parameters in the theory (in addition to those already present in the Standard model): (i) the mass of the new scalar $m_0$, (ii) dimensionless self coupling of the scalar $\lambda_S$, and (iii) a dimensionless coupling to the Higgs $\lambda$.
The power-counting renormalizable Lagrangian of the model, is therefore, $$\mathscr{L}=\mathscr{L}_{\rm SM}+\frac{1}{2}(\partial_\mu S)^2-\frac{1}{2}m_0^2S^2-\frac{1}{4}\lambda_SS^4-\lambda S^2(H^\dagger H)\tag{1}$$ where a $\mathbb{Z}_2$ invariance is proposed under which all the SM fields are even but S is odd i.e. $S\to -S$ under $\mathbb{Z}_2$. This forbids linear ($\lambda_1 S$), cubic ($\lambda_3S^3$), and $\lambda^\prime S(H^\dagger H)$ terms in the Lagrangian.
As I can understand, by forbidding the term $\sim S(H^\dagger H)$ "one can prevent the decay of the dark matter $S$ into a pair of SM Higgs bosons", and maintain the constancy of the observed cosmological relic abundance. But that will anyway be kinematically forbidden if the mass of $S$ turns out to be less than twice the mass of the Higgs.
Addendum If the term $S(H^\dagger H)$ is absent, how is the following annihilation $SS\to H\to XX$, where $X=g,b, W,Z^0$ (as depicted in Fig. 1) become possible? Am I missing something? Moreover, if it happens, will it not continually deplete the relic abundance?
Can someone enlighten me keeping in mind that I'm no expert in this field.