The Inductive Kernels of Graphs

TL;DR

This paper introduces inductive kernels in graphs, proving their universal existence and providing a quadratic-time construction method, while also analyzing the computational complexity of related decision problems.

Contribution

It presents a new inductive definition of kernels, establishes their guaranteed existence, and analyzes the complexity of related existence problems.

Findings

Inductive kernels always exist in graphs.

A particular inductive kernel can be constructed in quadratic time.

Deciding the existence of an inductive kernel with certain vertices is NP-Complete.

Abstract

It is well known that kernels in graphs are powerful and useful structures, for instance in the theory of games. However, a kernel does not always exist and Chv\'atal proved in 1973 that it is an NP-Complete problem to decide its existence. We present here an alternative definition of kernels that uses an inductive machinery : the inductive kernels. We prove that inductive kernels always exist and a particular one can be constructed in quadratic time. However, it is an NP-Complete problem to decide the existence of an inductive kernel including (resp. excluding) some fixed vertex.