(k,ℓ)-kernels in the generalized Mycielskian of digraphs

Abstract

Let [Formula: see text] and [Formula: see text] be positive integers. A set [Formula: see text] is said to be a [Formula: see text]-kernel of a digraph [Formula: see text] if for all [Formula: see text] [Formula: see text] and for every [Formula: see text] there exists [Formula: see text] such that [Formula: see text] A (2,1)-kernel is called a kernel. The generalized Mycielskian [Formula: see text] of a loopless digraph [Formula: see text] is defined in [S. F. Raj, Connectivity of the generalized Mycielskian of digraphs, Graphs Combin. 29 (2013) 893–900] as the digraph whose vertex set is [Formula: see text] where [Formula: see text] [Formula: see text] and arc set [Formula: see text] It is proved in [S. Vidhyapriya, Contributions to the theory of Kernels in digraphs, Doctoral Dissertation, Annamalai University, India (2016)] that [Formula: see text] the Mycielskian of [Formula: see text] has a kernel. In this paper, we generalize this result to [Formula: see text] Further, we completely study the structure of [Formula: see text]-kernels (if exist) in [Formula: see text] Besides these, for [Formula: see text] [Formula: see text] and [Formula: see text] we give some results concerning the existence and non-existence of [Formula: see text]-kernels in [Formula: see text] where, for some pairs [Formula: see text] we provide a Characterization, too.