Semikernels and ( k,l )-Kernels in Digraphs
Hortensia Galeana‐Sánchez, Xueliang Li · SIAM Journal on Discrete Mathematics · 1998
Let D be a digraph with minimum indegree at least one. The following results are proved: a digraph D has a semikernel if and only if its line digraph $L(D)$ does; the number of (k,1)-kernels in L(D) is less than or equal to that in D; if the number of (k,l)-kernels in D is less than or equal to the number of (2,l)-kernels in L(D), and if L(D) has a (k,l)-kernel, then D has a (k',l')-kernel for $k'+l\leq k$, $l\leq l'$. As a consequence, it obtains previous results about kernels and quasikernels in the line digraph. It is also proved that any digraph has a (k,l)-kernel with $l\geq 2k-2$, $k\geq 1$, generalizing a previous result on the existence of quasikernels in digraphs.