**Abstract:** A set *D* of vertices in a graph *G* is a *dominating set* if every vertex of *G* not in *D* has a neighbor in *D*. The size of a smallest dominating set, denoted *\gamma(G)*, is the *domination number* of *G*. We report on recent results with N. Ananchuen, with X. Zha, and with K. Kawarabayashi and A. Saito about three different topics involving domination in graphs.

A graph *G* is *\gamma-edge-critical* if *\gamma(G+e)<\gamma(G)* for each edge *e\notin E(G)*. It is *\gamma*-vertex-critical if *\gamma (G-v)<\gamma(G)* for every vertex *v\in V(G)*. The structure of *\gamma*-edge-critical graphs and *\gamma*-vertex-critical graphs is not well understood, even when *\gamma(G)=3*. We present new results on both classes that involve matchings.

In 1996, Matheson and Tarjan proved that a triangulated disc with *n* vertices has domination number at most *n/3*, and thus so does every *n*-vertex triangulation. We will present recent work toward extending this result to graphs of higher genus.

Reed conjectured in 1996 that if *G* is a cubic graph with *n* vertices, then *\gamma (G) <= \lceil|V(G)|/3\rceil*. This conjecture was very recently shown to be false by Kostochka and Stodolsky. We will close by discussing new results pertaining to this conjecture.