Characterizations for split graphs and unbalanced split 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 $\alpha (G)\ge 3$ 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.