Saturated simple and k-simple topological graphs

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered.Comment: 25 pages, 17 figures, added some new results and improvement

Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algebras, $\mathcal{O}(E,C)$ and $\mathcal{O}^r(E,C)$, of a separated graph $(E,C)$, and of the abelianized Leavitt path algebra $L_K^{\text{ab}}(E,C)$ as well. These algebras are the (reduced) crossed products with respect to the above-mentioned partial actions, and we prove that there is a lattice isomorphism between the lattice of induced ideals and the lattice of hereditary $D^{\infty}$-saturated subsets of a certain infinite separated graph $(F_{\infty},D^{\infty})$ built from $(E,C)$, called the separated Bratteli diagram of $(E,C)$. We finally use these tools to study simplicity and primeness of the tame separated graph algebras.Comment: 60 page