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Optimally reconnecting graphs against an edge-destroying adversary

Daniel C. McDonald

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.

Optimally reconnecting graphs against an edge-destroying adversary

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.

Paper Structure

This paper contains 15 sections, 26 theorems, 5 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Description of Model
  3. Example Gameplay
  4. Statement of Main Theorem
  5. Related Work
  6. Avenues for Future Research
  7. Facts about Fixer-superiority and Fixer-dominance
  8. Proof of Main Theorem
  9. The case that $G^{\mathbb{h}}_{1}-B^{\mathbb{h}}_{1}$ has one component
  10. The case that $G^{\mathbb{h}}_{1}-B^{\mathbb{h}}_{1}$ has two components
  11. The case that $G^{\mathbb{h}}_{1}-B^{\mathbb{h}}_{1}$ has at least three components
  12. Details of Proof of Proposition \ref{['singStrat']}
  13. Details of Proof of Proposition \ref{['subsetSupProp']}
  14. Details of Proof of Proposition \ref{['proofExecution']}
  15. Details of Proof of Lemma \ref{['equivalenceLem']}

Key Result

Theorem 1.1

Any greedy Fixer strategy is Fixer-dominant at any history $\mathbb{h}$.

Figures (8)

  • Figure 1: The solid edges form the graph $G$, while the dashed edges form the reserve set $R$, with $w(e_4)=1$ and $w(e_5)=2$.
  • Figure 2: A decision tree for $\phi$, where $\phi(h)=\{e_4\}$.
  • Figure 3: A decision tree for $\phi'$, where $\phi'(h)=\{e_5\}$. Observe that the rest of the decision tree is forced. Indeed, $G_{2}=\{e_3,e_5\}$, which is a path graph, and $R_{2}=\{e_4\}$, where $e_4$ spans the endpoints of that path. Thus Buster deleting any single edge from $G_{2}$ necessitates Fixer reconnecting the graph using $e_4$, from which point on any further deletions by Buster cannot be countered by Fixer, and Buster initially deleting both edges from $G_{2}$ also cannot be countered by Fixer.
  • Figure 4: A decision tree for $\phi"$, where $\phi"(h)=\{e_4,e_5\}$. Again, observe that the rest of the decision tree is forced. Indeed, $R_{2}$ is empty, leaving Fixer with no options besides playing the empty set when possible.
  • Figure 5: Graphs $G^{\mathbb{h}[{\beta^1,\phi^1}]}_{k}$ and $G^{\mathbb{h}[{\beta^g,\phi^g}]}_{k}$ from Scenario \ref{['sce1']}.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 41 more