A graph $G$ of order $2n$ is called degree-equipartite if for every $n$-element set $A\subseteq V(G)$, the degree sequences of the induced subgraphs $G[A]$ and $G[V(G)\setminus A]$ are the same. In this paper, we characterize all degree-equipartite graphs. This answers Problem 1 in the paper by Gr\"{ u}nbaum et al [B. Gr\"{ u}nbaum, T. Kaiser, D. Kr\'{ a}l, and M. Rosenfeld, Equipartite graphs, { \it Israel J. Math.} { \bf 168} (2008), 431-444].
Categories: math.CO