The 2-domination number is at least the radius

Let G be a connected graph. A set S of vertices is 2-dominating if everyvertex outside S has at least two neighbors in S, and gamma_2(G) is theminimum size of such a set. We prove that gamma_2(G) >= rad(G). In fact weprove the stronger statement that every 2-dominating set S admits a vertex cwith d(c,x) <= |S|-1 for all x in S, so that rad(G) <= |S|. The bound issharp: even cycles satisfy gamma_2 = rad, and we characterize the extremalgraphs of radius at least 3 (the "blob cycles"). The proof introduces a portabsorption construction, which builds from a 2-dominating set a spanning treecontaining it whose radius is at most |S|-1. Preprint. To the best of our knowledge no lower bound on gamma_2 in terms ofthe radius has previously appeared; this claim awaits independent humanrefereeing. Results were found with the aid of an automatedconjecture-and-verification system and every proof has since been written andaudited by hand.