The 2-Domination Number and the Upper Median Degree: A Proof of Graffiti.pc Conjecture 387

Abstract

Let G be a nonempty finite simple graph of order n, and let m(G) be the upper median of its degree sequence. We prove that the 2-domination number satisfies gamma_2(G)<= n - m(G) + 1. This proves Graffiti.pc Conjecture 387. In fact, the argument establishes the inequality for every nonempty finite simple graph, so the connectedness hypothesis in the original formulation is unnecessary. The proof uses the complement graph and a minimally linearly dependent family of polynomials encoding selected nonneighborhoods.