Computer Science / Discrete Mathematics Mathematics / math.CO

Revolutionaries and spies on trees and unicyclic graphs

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📝 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 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Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael 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 Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael 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 Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael Smyth, Michael Howard, Michael 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 Smyth, Michael Howard, Michael Smyt

📝 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

Figure 1

📄 Full Content

Many pursuit games have been studied on graphs. We study such a game that can be interpreted as modeling a problem of network security. One team consists of r revolutionaries; the other consists of s spies. The revolutionaries want to arrange a one-time meeting of m revolutionaries free of oversight by spies. Initially, the revolutionaries take positions at vertices, and then the spies do the same. In each subsequent round, each revolutionary may move to an adjacent vertex or not move, and then each spy has the same option. Everyone knows where everyone else is.

The revolutionaries win if at the end of a round there is an unguarded meeting, where a meeting is a set of (at least) m revolutionaries on one vertex, and a meeting is unguarded if there is no spy at that vertex. The spies win if they can prevent this forever. Let RS(G, m, r, s) denote this game played on the graph G by s spies and r revolutionaries seeking an unguarded meeting of size m.

The revolutionaries can form min{|V (G)|, ⌊r/m⌋} meetings initially; if s is smaller than this, then the spies immediately lose. On the other hand, the spies win if s ≥ r -m + 1; they follow r -m + 1 distinct revolutionaries, and the other m -1 revolutionaries cannot form a meeting. For fixed G, r, m we study the minimum s such that the spies win RS(G, m, r, s); let σ(G, m, r) denote this threshold. The trivial bounds are min{|V (G)|, ⌊r/m⌋} ≤ σ(G, m, r) ≤ min{|V (G)|, r -m + 1}.

The game was invented by Jozef Beck in the mid-1990s. Shortly thereafter, Smyth proved that the trivial lower bound on σ(G, m, r) is sufficient when G is a tree. This was not published; we include a proof here and use the result in solving the game for unicyclic graphs. Howard and Smyth [2] studied RS(G, 2, r, s) when G is the infinite 2-dimensional integer grid with one-step horizontal, vertical, and diagonal edges. They proved 6 ⌊r/8⌋ ≤ σ(G, 2, r) ≤ r -2. For the upper bound, which they conjectured is sharp, they showed that r -2 spies can win by having all but one follow revolutionaries; the one remaining spy can prevent the remaining three revolutionaries from making an unguarded meeting.

In this paper, we determine σ(G, m, r) for all trees and unicyclic graphs. For a graph G with r/m < |V (G)|, we show that σ(G, m, r) ≤ ⌈r/m⌉ if G has at most one cycle, by summing over components. When G is a tree, ⌊r/m⌋ spies suffice, and ⌈r/m⌉ suffice when G is connected and unicyclic. Our final result is that if G is unicyclic with a cycle of length ℓ and t vertices not on the cycle (connected or not), and m ∤ r, then σ(G, m, r) = ⌊r/m⌋ if and only if ℓ ≤ max{⌊r/m⌋ -t + 2, 3}.

Butterfield, Cranston, Puleo, West, and Zamani [1] study σ(G, m, r) on a variety of graphs. With r/m < |V (G)|, they show that σ(G, m, r) = ⌊r/m⌋ when G has a rooted spanning tree T such that every edge of G not in T joins vertices having the same parent in T ; this includes graphs with a dominating vertex and interval graph. For every graph G, they prove σ(G, m, r) ≤ γ(G) ⌊r/m⌋, where γ(G) is the domination number of G, and this is nearly sharp: for t, m, r ∈ N with t ≤ m, there is a graph G with domination number t such that σ(G, m, r) > t(r/m -1). Also, there are chordal graphs (and bipartite graphs) for given r and m such that σ(G, m, r

They also consider r-large complete k-partite graphs, where “r-large” means that each partite set has size at least 2r. When k ≥ m and G is such a graph,

When G is a large bipartite graph, they show that σ(G, 2, r) = 7r/10 and σ(G, 3, r) = 3r/2 (within additive constants) and that in general

The upper bound coefficient on r m is about 1.58; the lower one is about 1.5 when m is large and r > m; they conjecture that the lower one is the true coefficient, at least when 3 | m.

If r/m < |V (G)|, then the spies lose whenever s < ⌊r/m⌋. Hence the first chance for the spies is when s = ⌊r/m⌋. We prove that this suffices when G is a tree, yielding σ(G, m, r) = ⌊r/m⌋. Since the spies always win when s ≥ |V (G)|, the statement of our first theorem remains true regardless of the relationship between r/m and |V (G)|. Nevertheless, to avoid trivial statements, we henceforth assume always that r/m ≤ |V (G)|.

Theorem 1. If G is a tree and s ≥ ⌊r/m⌋, then the spies win RS(G, m, r, s).

Proof. It suffices to show that the spies win when s = ⌊r/m⌋ < |V (G)|. Choose a root z ∈ V (G). The parent of a non-root vertex v, denoted v + , is its neighbor on the path from v to z. The other neighbors of v are its children; let C(v) be this set of children of V . The descendants of v are all the vertices (including v) whose path to z contains v; let D(v) be the set of descendants of v.

For each vertex v, let r(v) and s(v) denote the number of revolutionaries and spies on v at the current time, respectively, and let w(v) = u∈D(v) r(u). The spies maintain the following invariant giving the number of spies on each vertex at the end of any round:

Since u∈C(v) w(u) = w(v) -r(v), the formula is always nonnegativ

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