# Semi-Grundy function, an hereditary approach to Grundy function

## TL;DR

This paper introduces the semi-Grundy function, an hereditary concept related to Grundy functions in digraphs, providing new insights into their existence, properties, and bounds, with implications for game theory and graph coloring.

## Contribution

It defines the semi-Grundy function, explores its relationship with Grundy functions, and establishes conditions and bounds for their existence and size in digraphs.

## Key findings

- Semi-Grundy functions are related to kernels and Grundy functions in digraphs.
- Conditions are provided for the existence of semi-Grundy functions in product graphs.
- Bounds are established for the size of semi-Grundy functions, impacting chromatic number estimates.

## 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 $n\geq 2$, there is a digraph $R_n$ 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.