  We introduce and study the "burning-and-sticking" (BS) lattice model for the
porous material that involves sticking of emerging finite clusters to the
mainland. In contrast with other single-cluster models, it does not demonstrate
any phase transition: the backbone exists at arbitrarily low concentrations.
The same is true for hybrid models, where the sticking events occur with
probability $q$: the backbone survives at arbitrarily low $q$. Disappearance of
the phase transition is attributed to the backbone reinforcement effect,
generic for models with sticking. A relation between BS and the cluster-cluster
aggregation is briefly discussed.
