On P4-transversals of perfect graphs Original Research Article

A subset T of vertices of a graph G is called a P4-transversal if T meets every P4 of G; if in addition T induces a P4-free subgraph then T is called a two-sided P4-transversal of G. We conjecture that graphs containing a two-sided P4-transversal are perfect provided they contain no odd chordless cycle with at least five vertices or the complement of such a cycle. We show that, as a consequence of previously known results, this conjecture is true for graphs containing a stable P4-transversal. We show, using a reduction from 3-SAT, that it is NP-complete to decide if a perfect graph has a stable P4-transversal. Next, we prove that a graph is perfectly orderable, in the sense of Chvátal, if it has a stable set meeting all P4ʹs in an end-point or all P4ʹs in a mid-point. We show that there is a polynomial time algorithm to recognize such a graph using a reduction to 2-SAT. Finally, we prove that our conjecture is true if the P4-transversal induces a threshold graph and meets every P4 in an end-point, or meets every P4 in a mid-point.