  We consider a mortal random walker on a family of hierarchical graphs in the
presence of some trap sites. The configuration comprising the graph, the
starting point of the walk, and the locations of the trap sites is taken to be
exactly self-similar as one goes from one generation of the family to the next.
Under these circumstances, the total probability that the walker hits a trap is
determined exactly as a function of the single-step survival probability $q$ of
the mortal walker. On the $n^{\rm th}$ generation graph of the family, this
probability is shown to be given by the $n^{\rm th}$ iterate of a certain
scaling function or map $q \rightarrow f(q)$. The properties of the map then
determine, in each case, the behavior of the trapping probability, the mean
time to trapping, the temporal scaling factor governing the random walk
dimension on the graph, and other related properties. The formalism is
illustrated for the cases of a linear hierarchical lattice and the Sierpinski
graphs in $2$ and $3$ Euclidean dimensions. We find an effective reduction of
the random walk dimensionality due to the ballistic behavior of the surviving
particles induced by the mortality constraint. The relevance of this finding
for experiments involving travel times of particles in diffusion-decay systems
is discussed.
