  Let G be a simple (i.e., no loops and no multiple edges) graph. We
investigate the question of how to modify G combinatorially to obtain a
sequentially Cohen-Macaulay graph. We focus on modifications given by 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 in G. We give various
sufficient conditions and necessary conditions on a subset S of the vertices of
G so that the graph G \cup 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 \cup W(S) is a sequentially Cohen-Macaulay
graph. On the other hand, we show that if G \backslash S is not sequentially
Cohen-Macaulay, then G \cup 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.
