Whiskers and sequentially Cohen–Macaulay graphs

We investigate how to modify a simple graph G combinatorially to obtain a sequentially Cohen–Macaulay graph. We focus on adding configurations of whiskers to G, where to add a whisker one adds a new vertex and an edge connecting this vertex to an existing vertex of G. We give various sufficient conditions and necessary conditions on a subset S of the vertices of G so that the graph G∪W(S), obtained from G by adding a whisker to each vertex in S, is a sequentially Cohen–Macaulay graph. For instance, we show that if S is a vertex cover of G, then G∪W(S) is a sequentially Cohen–Macaulay graph. On the other hand, we show that if G∖S is not sequentially Cohen–Macaulay, then G∪W(S) is not a sequentially Cohen–Macaulay graph. Our work is inspired by and generalizes a result of Villarreal on the use of whiskers to get Cohen–Macaulay graphs.