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

TL;DR

This paper investigates graphs with specific degree sequences related to disjoint cliques and their complements, analyzing their structural properties and computational complexity of related optimization problems.

Contribution

It characterizes properties of graphs with degree sequences of two disjoint cliques and their complements, and proves NP-hardness of classical optimization problems on these graphs.

Findings

Graphs with degree sequences $ extstyleiglrace m^{m-1},n^{n-1}igr brace$ and $ extstyleiglrace m^n,n^m igr brace$ have specific connectivity and Hamiltonian properties.

Recognition and optimization problems on these graphs are NP-hard.

Abstract

In this paper we study the class of graphs $G_{m,n}$ that have the same degree 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 various properties of $G_{m,n}$ and $\overline G_{m,n}$ related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard.