In my studies of graph theory I recently came across the following:
Let $ G $ be a finite simple undirected graph which is chordal and for some $ e \in E(G) $ the graph $ G-e $ is also chordal then there exists a unique maximal clique $ K $ in $ G $ such that both ends of $ e $ are in $ K $.
A finite simple undirected graph is chordal if every cycle of length greater than three has a chord . Equivalently, a graph is chordal if it has no induced cycle of length greater than three.
A useful theorem I thought might help here
A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering.
I cannot seem to relate the graph being chordal and the modified graph being chordal to the existence of any clique, let alone to a unique one with both ends of $ e $ are in this clique. I thought perhaps perfect elimination order would be a clue here but I cannot proceed any further. I certainly appreciate all help.