Semi-Grundy function, an hereditary approach to Grundy function
Abstract
Grundy functions have found many applications in a wide variety of games, in solving relevant problems in Game Theory. Many authors have been working on this topic for over many years. Since the existence of a Grundy function on a digraph implies that it must have a kernel, the problem of deciding if a digraph has a Grundy function is NP-complete, and how to calculate one is not clearly answered. In this paper, we introduce the concept: Semi-Grundy function, which arises naturally from the connection between kernel and semi-kernel and the connection between kernel and Grundy function. We explore the relationship of this concept with the Grundy function, proving that for digraphs with a defining hereditary property is sufficient to get a semi-grundy function to obtain a Grundy function. Then we prove sufficient and necessary conditions for some products of digraphs to have a semi-Grundy function. Also, it is shown a relationship between the size of the semi-Grundy function obtained for the Cartesian Product and the size of the semi-Grundy functions of the factors. This size is an upper bound of the chromatic number. We present a family of digraphs with the following property: for each natural number n≥ 2, there is a digraph Rn that has two Grundy functions such that the difference between their maximum values is equal to n. Then it is important to have bounds for the Grundy or semi-Grundy functions.