Open Problems #14

In Open Problems #3 (Summer 1991), I mentioned several analogues or extensions of Edmonds' Branching Theorem. Let 0 (G; r) be the minimum number of edges that must be deleted to make some vertex of a digraph G unreachable from r, and let (G; r) be the analogous parameter for vertex deletion. Edmonds proved that G has 0 (G; r) edge-disjoint r-branchings in G, where an r-branching is a subgraph in which r has in-degree 0 and every other vertex has in-degree 1. Andrr as Frank conjectured a vertex analogue, in which 0 is replaced by and \edge-disjoint" is replaced by \internally-disjoint". Nate Dean reports that Andreas Houck 16] discovered a counterexample to this conjecture. As far as I know, the related conjectures in OP#3 by K ezdy (a strengthening of Edmonds result for digraphs) and by Itai and Rodeh (analogous statements for undi-rected graphs) remain open. Open Problems #6 (Spring 1992) included Steve Steinsaltz's question of which posets arise as the poset C(G) of connected induced subgraphs of a connected graph G. K ezdy and Seif 22] [email protected] have proved that a poset P is C(G) for some connected graph G if and only if P is an atom-height, properly semimod-ular poset. Letting A(x) denote the set of atoms (height-one elements) below x, an atom-height poset is a ranked poset having a unique minimal element 0, in which every element x is a least upper bound of A(x) and has height jA(x)j. It was known that C(G) is a lattice if and only if G is a Husimi tree (every block is a clique); Jacobson, K ezdy, and Seif 19] have proved that the nite lattices arising in this way are the atomic lattices that are dually locally distributive and properly distributive. A lattice is atomic if each element is a join of atoms. A lattice is dually locally distributive if for each element b, the interval a; b] is distributive, where a is the meet of the set of elements covered by b. Open Problems #12 (Fall 1993) included Steve Penrice's questions about the Sperner-theoretic properties of the posets C(G). Jacobson, K ezdy, and Seif 18] have obtained trees T for which the poset C(T) is not Sperner. Penrice's questions relating C(G) and the LYM property remain open; what is the minimum number of edges in G that forces C(G) to be LYM when G has n vertices, …