Edge erasures and chordal graphs

Jared Culbertson (Sensors Directorate, Air Force Research Laboratory, 2241 Avionics Circle, Bldg. 620, Wright--Patterson Air Force Base, Ohio 45433-7302, USA)
Dan P. Guralnik (Mechanical & Aerospace Engineering, University of Florida, MAE-B, room 327A, 633 Gale Lemerand Dr, Gainesville, FL 32603, USA.)
Peter F. Stiller (Department of Mathematics, MS3368, Texas A&M University, College Station, Texas 77843-3368, USA.)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications

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.