  We prove several results about chordal graphs and weighted chordal graphs by
focusing on exposed edges. These are edges that are properly contained in a
single maximal complete subgraph. This leads to a characterization of chordal
graphs via deletions of a sequence of exposed edges from a complete graph. Most
interesting is that in this context the connected components of the
edge-induced subgraph of exposed edges are 2-edge connected. We use this latter
fact in the weighted case to give a modified version of Kruskal's second
algorithm for finding a minimum spanning tree in a weighted chordal graph. This
modified algorithm benefits from being local in an important sense.
