Optimally reconnecting graphs against an edge-destroying adversary

TL;DR

This paper models a game where two adversaries alternately modify a connected graph, analyzing optimal strategies for reconnecting the graph against edge deletions with cost considerations.

Contribution

It introduces a new game-theoretic model for graph reconnection with weighted edges and proves the optimality of a greedy strategy for Fixer.

Findings

Greedy strategy for Fixer is always optimal.

Fixer can guarantee victory or the minimal cost of defeat.

The model accounts for weighted edges and adversarial deletions.

Abstract

We introduce a model involving two adversaries Buster and Fixer taking turns modifying a connected graph, where each round consists of Buster deleting a subset of edges and Fixer responding by adding edges from a finite reserve set of weighted edges to leave the graph connected, with Buster limited by the total number of edges he is allowed to delete throughout the game. Fixer wins if she can reconnect the graph after Buster has reached his limit of edges to delete, while Buster wins if he can delete edges in such a way that Fixer cannot reconnect the graph using the remaining edges in reserve. With the weights representing the cost for Fixer to use specific reserve edges to reconnect the graph, we prove that a greedy strategy for Fixer always results in an optimal result for Fixer: victory, if possible, for as cheaply as can be guaranteed against any Buster strategy, and if defeat cannot be avoided, the cheapest possible loss that can be guaranteed against any Buster strategy.