 
Summary: 1. If v is a pendant vertex in a graph G, then rv(G) = 0 or 1. Can we
characterize when each case occurs?
2. For which graphs is rv(G) = 2 for all v V (G)?
3. Can we characterize which vertices in a graph have rv(G) = 2?
Let G be a graph with a cutvertex v. Then rv(G) = 2
(or v is rank strong) if and only if one of the following (or
potentially both) occur:
(a) rv(H) = 2 for H a subgraph of G with the property that
H  v is a component of G  v.
(b) There exist two subgraphs H1 and H2 of G with the
property that H1 v and H2 v are distinct components
of G  v and rv(H1) = 1 and rv(H2) = 1.
The proof of this is from the equation
mr(G) = mr(Hi  v) + min{2, rv(Hi)}, (0.1)
where the sum is taken over all components Hiv of Gv (so
Hi is a subgraph of G that contains v and has the property
that Hi  v is a component of G  v).
This is only a partial answer, since we know there are
graphs with rank strong vertices that are not cut vertices.
4. (a) For which graphs is Z(G) = M(G)?
