  We study the joint occurrence of large values of a Markov random field or
undirected graphical model associated to a block graph. On such graphs,
containing trees as special cases, we aim to generalize recent results for
extremes of Markov trees. Every pair of nodes in a block graph is connected by
a unique shortest path. These paths are shown to determine the limiting
distribution of the properly rescaled random field given that a fixed variable
exceeds a high threshold. The latter limit relation implies that the random
field is multivariate regularly varying and it determines the max-stable
distribution to which component-wise maxima of independent random samples from
the field are attracted. When the sub-vectors induced by the blocks have
certain limits parametrized by H\"usler-Reiss distributions, the global Markov
property of the original field induces a particular structure on the parameter
matrix of the limiting max-stable H\"usler-Reiss distribution. The multivariate
Pareto version of the latter turns out to be an extremal graphical model
according to the original block graph. Thanks to these algebraic relations, the
parameters are still identifiable even if some variables are latent.
