Graffiti . pc on the k-independence number of a graph

The k-independence number of a graph is the cardinality of a largest set of vertices that induce a subgraph of maximum degree at most k − 1. We prove several conjectures made by the computer program Graffiti.pc and present several of the remaining open conjectures. keywords: k-independence number, independence number, WelshPowell, Graffiti.pc, neighbor dominators, induced subgraph. 1 Definitions and Introduction Given a finite simple graph G = (V, E), an independent set is a subset of V such that no pair of vertices in the subset are adjacent. The cardinality of a maximum independent set is called the independence number of G and is denoted by β(G). For a positive integer k, a k-independent set is a subset Ik of V such that the subgraph induced by Ik has degree at most k−1. The cardinality of a maximum k-independent set is called the k-independence number of G and is denoted by βk(G). Note, that k-independent sets are ∗Work supported in part by the United States Department of Defense and resources of the Extreme Scale Systems Center at Oak Ridge National Laboratory.