Revolutionaries and spies on trees and unicyclic graphs

📝 Original Info

• Title: Revolutionaries and spies on trees and unicyclic graphs
• ArXiv ID: 1110.2274
• Date: 2023-11-02
• Authors: Jozef Beck, Michael Smyth, Howard, William Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, 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Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael 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Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael 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Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, 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Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, 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📝 Abstract

A team of $r$ {\it revolutionaries} and a team of $s$ {\it spies} play a game on a graph $G$. Initially, revolutionaries and then spies take positions at vertices. In each subsequent round, each revolutionary may move to an adjacent vertex or not move, and then each spy has the same option. The revolutionaries want to hold an {\it unguarded meeting}, meaning $m$ revolutionaries at some vertex having no spy at the end of a round. To prevent this forever, trivially at least $\min\{|V(G)|,\FL{r/m}\}$ spies are needed. When $G$ is a tree, this many spies suffices. When $G$ is a unicyclic graph, $\min\{|V(G)|,\CL{r/m}\}$ spies suffice, and we characterize those unicyclic graphs where $\FL{r/m}+1$ spies are needed. \def\FL#1{\lfloor #1 \rfloor} \def\CL#1{\lceil #1 \rceil}

💡 Deep Analysis

Deep Dive into Revolutionaries and spies on trees and unicyclic graphs.

A team of $r$ {\it revolutionaries} and a team of $s$ {\it spies} play a game on a graph $G$. Initially, revolutionaries and then spies take positions at vertices. In each subsequent round, each revolutionary may move to an adjacent vertex or not move, and then each spy has the same option. The revolutionaries want to hold an {\it unguarded meeting}, meaning $m$ revolutionaries at some vertex having no spy at the end of a round. To prevent this forever, trivially at least $\min\{|V(G)|,\FL{r/m}\}$ spies are needed. When $G$ is a tree, this many spies suffices. When $G$ is a unicyclic graph, $\min\{|V(G)|,\CL{r/m}\}$ spies suffice, and we characterize those unicyclic graphs where $\FL{r/m}+1$ spies are needed. \def\FL#1{\lfloor #1 \rfloor} \def\CL#1{\lceil #1 \rceil}

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