Nordhaus–Gaddum-Type Results for the k-Independent Number of Graphs

Abstract

The concept of [Formula: see text]-independent set, introduced by Fink and Jacobson in 1986, is a natural generalization of classical independence set. A [Formula: see text]-independent set is a set of vertices whose induced subgraph has maximum degree at most [Formula: see text]. The [Formula: see text]-independence number of [Formula: see text], denoted by [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-independent set of [Formula: see text]. As a natural counterpart of the [Formula: see text]-independence number, we introduced the concept of [Formula: see text]-edge-independence number. An edge set [Formula: see text] in [Formula: see text] is called [Formula: see text]-edge-independent if the maximum degree of the subgraph induced by the edges in [Formula: see text] is less or equal to [Formula: see text]. The [Formula: see text]-edge-independence number, denoted [Formula: see text], is defined as the maximum cardinality of a [Formula: see text]-edge-independent set. In this paper, we study the Nordhaus–Gaddum-type results for the parameter [Formula: see text] and [Formula: see text]. We obtain sharp upper and lower bounds of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. Some graph classes attaining these bounds are also given.