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In this paper we study the class of graphs $G_{m,n}$ that have the samedegree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class$\overline G_{m,n}$ of the complements of such graphs. We establish variousproperties of $G_{m,n}$ and $\overline G_{m,n}$ related to recognition,connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We alsoshow that several classical optimization problems on these graphs are NP-hard.

Graphs with degree sequence $\{m^{m-1},n^{n-1}\}$ and $\{m^n,n^m\}$