Cohen-Macaulay Growing Graphs

We introduce a new family of simple graphs, so called, growing graphs. We investigate ways to modify a given simple graph G combinatorially to obtain a growing graph. One may obtain infinitely many growing graphs from a single simple graph. We show that a growing graph obtained from any given simple graph is Cohen–Macaulay and every Cohen–Macaulay chordal graph is a growing graph. We also prove that under certain conditions, a graph is growing if and only if its clique complex is grafted and give several equivalent conditions in this case. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers and the work of Faridi on grafting of simplicial complexes.