A set of vertices S in a graph G dominates G if every vertex of G either belongs to S or is a neighbor of a member of S. The size of any smallest dominating set in G is called the domination number of G and is denoted by gamma(G).

We will discuss recent progress in several areas of graph domination:

(a) The structural and matching properties of gamma-edge-critical graphs. Graph G is gamma-edge-critical if gamma(G + e) < gamma(G), for every edge e in the complement of G.

(b) The structural and matching properties of gamma-vertex-critical graphs. Graph G is gamma-vertex-critical if gamma(G - v) < gamma(G), for every vertex v in G.

(c) Reed's domination conjecture for cubic graphs. In 1996, Reed proved that every graph G of minimum degree at least three satisfies gamma(G) leq (3/8) |V (G)| and conjectured that if G is a connected cubic graph, then gamma(G) leq ceil( |V (G)/3| ).