Characterizations for split graphs and unbalanced split graphs

TL;DR

This paper introduces new characterizations of split graphs and related classes using edge contraction, providing insights into their structure and connections to other graph classes like Nordhaus-Gaddum graphs.

Contribution

It presents novel characterizations of split graphs and unbalanced split graphs via edge contraction, and applies these to classify (2K2, claw)-free graphs and Nordhaus-Gaddum graphs.

Findings

Any (2K2, claw)-free graph with independence number ≥ 3 is a split graph

Characterization of pseudo-split graphs using edge contraction

Unbalanced split graphs are characterized and linked to Nordhaus-Gaddum graphs

Abstract

We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $\alpha(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph. Finally, by using edge contraction again, we characterize unbalanced split graphs which we use to characterize the Nordhaus-Gaddum graphs.