Kernel-Perfection through the Push Operation

Let $D = (V,A)$ be a digraph.A kernel of $D$ is an independent set $S$ of vertices such that every vertexof $D$ is either in $S$ or dominates a vertex in $S$. If every inducedsubdigraph of $D$ has a kernel, then $D$ is called kernel-perfect.According to Richardson, if a digraph does not contain a directed cycle of oddlength then it is kernel-perfect. Here we study the kernel-perfection throughuse of the push operation of digraphs. For a subset $X$ of vertices of $D$,$D^X$ is the digraph obtained from $D$ by pushing $X$, that is, reversingthe directions of arcs between $X$ and $V-X$. We prove that the problem ofdeciding if a digraph can be pushed to be kernel-perfect is an NP-completeproblem. This is on contrast to a previous result showingthe same decision problem restricted to chordal digraphs is polynomial timesolvable. We further show that the problem of deciding whether a graphcan be pushed to contain no directed cycle of odd length is also NP-complete.