1. If v is a pendant vertex in a graph G, then rv(G) = 0 or 1. Can we characterize when each case occurs? 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 cut-vertex 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 Hi-v of G-v (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)? Collections: Mathematics