  The three subgraphs of a connected graph induced by the center, annulus and
periphery are called its metric subgraphs. The main results are as follows. (1)
There exists a graph of order $n$ whose metric subgraphs are all paths if and
only if $n\ge 13$ and the smallest size of such a graph of order $13$ is $22;$
(2) there exists a graph of order $n$ whose metric subgraphs are all cycles if
and only if $n\ge 15,$ and there are exactly three such graphs of order $15;$
(3) for every integer $k\ge 3,$ we determine the possible orders for the
existence of a graph whose metric subgraphs are all connected $k$-regular
graphs; (4) there exists a graph of order $n$ whose metric subgraphs are
connected and pairwise isomorphic if and only if $n\ge 24$ and $n$ is divisible
by $3.$ An unsolved problem is posed.
