Inheritance of Convexity for the $\mathcal{P}_{\min}$-Restricted Game

TL;DR

This paper characterizes and efficiently recognizes graphs that preserve convexity in restricted cooperative games based on minimum partitions induced by edge deletions.

Contribution

It provides a characterization and polynomial-time recognition algorithm for graphs that inherit convexity in Pmin-restricted games.

Findings

Graphs with inheritance of convexity are characterized.

Recognition of such graphs can be done in polynomial time.

The study extends understanding of convexity inheritance in cooperative game theory.

Abstract

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the subcomponents corresponding to a minimum partition. This minimum partition $\mathcal{P}_{\min}$ is induced by the deletion of the minimum weight edges. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with Pmin. Moreover, we prove that these graphs can be recognized in polynomial time.