Semikernels and (k,l)-Kernels in Digraphs

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.

  • [1]  Google Scholar

  • [2]  C. Berge and  and A. Ramachandra Rao, A combinatorial problem in logic, Discrete Math., 17 (1977), pp. 23–26. dsm DSMHA4 0012-365X Discrete Math. CrossrefISIGoogle Scholar

  • [3]  Google Scholar

  • [4]  P. Duchet, A sufficient condition for a digraph to be kernel‐perfect, J. Graph Theory, 11 (1987), 81–85 88c:05059 CrossrefISIGoogle Scholar

  • [5]  Pierre Duchet and , Henri Meyniel, Une généralisation du théorème de Richardson sur l’existence de noyaux dans les graphes orientés, Discrete Math., 43 (1983), 21–27 10.1016/0012-365X(83)90017-1 84g:05067 CrossrefISIGoogle Scholar

  • [6]  Pierre Duchet, Yahya Hamidoune and , Henry Meyniel, Sur les quasi‐noyaux d’un graphe, Discrete Math., 65 (1987), 231–235 10.1016/0012-365X(87)90054-9 88k:05161 CrossrefISIGoogle Scholar

  • [7]  H. Galeana‐Sánchez and , V. Neumann Lara, On kernels and semikernels of digraphs, Discrete Math., 48 (1984), 67–76 85i:05115 CrossrefISIGoogle Scholar

  • [8]  H. Galeana‐Sánchez, L. Pastrana Ramírez and , H. Rincón‐Mejía, Semikernels, quasi kernels, and Grundy functions in the line digraph, SIAM J. Discrete Math., 4 (1991), 80–83 92c:05068 LinkISIGoogle Scholar

  • [9]  Matúš Harminc, Solutions and kernels of a directed graph, Math. Slovaca, 32 (1982), 263–267 84h:05057 Google Scholar

  • [10]  M. Kwaśnik, The generalization of Richardson theorem, Discussiones Math., IV (1981), pp. 11–14. Google Scholar

  • [11]  Google Scholar

  • [12]  V. Neumann‐Lara, Seminúcleos en una digráfica, Anales del Instituto de Matemáticas de la Universidad Nacional Autónoma de México, México, 11 (1971), pp. 55–62. Google Scholar

  • [13]  M. Richardson, Solution of irreflective relations, Ann. Math., 58 (1953), pp. 573–580. ann ANMAAH 0003-486X Ann. Math. CrossrefISIGoogle Scholar