Contraction blockers for graphs with forbidden induced paths.

We consider the following problem: can a certain graph parameter of some given graph be reduced by at least d for some integer d via at most k edge contractions for some given integer k? We examine three graph parameters: the chromatic number, clique number and independence number. For each of these graph parameters we show that, when d is part of the input, this problem is polynomial-time solvable on P4-free graphs and NP-complete as well as W[1]-hard, with parameter d, for split graphs. As split graphs form a subclass of P5-free graphs, both results together give a complete complexity classification for Pℓ-free graphs. The W[1]-hardness result implies that it is unlikely that the problem is fixed-parameter tractable for split graphs with parameter d. But we do show, on the positive side, that the problem is polynomial-time solvable, for each parameter, on split graphs if d is fixed, i.e., not part of the input. We also initiate a study into other subclasses of perfect graphs, namely cobipartite graphs and interval graphs

Characterizations for split graphs and unbalanced split graphs

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