令 G = (V, E)為一個相連圖。我們說一個集合 U 為 P3-convex: 如果對於每個掉在 V \ U 的點，該點的鄰居最多只有一個點屬於 U。對於任意集合 W，我們用 σ(W)定義包含 W 的唯一且最小 P3-convex 集合。 兩個玩家在一個圖上玩 connected P3-game，輪流挑選沒有被標記過的點，遊戲開始所有點都沒有被標記，M 為被標記的點集合，一開始是空集合，玩家的每一步可以標記一個先前沒有被標記的點 v，使得 M
被更新為 M’ = σ(M ∪ {v})，且如果 M’是相連的，這步才可以下，
當該某為玩家時，如果發現所有點都被標記，則該玩家就輸了。在這
篇論文中，我們考慮在格子圖與圓環圖上玩 connected P3-game，我們
給了該遊戲的 Grundy 值公式。我們還進一步考慮在路徑和環上玩
connected Pr-game，當 r 大於等於 3。

Let G = (V,E) be a connected graph. We say a set U ⊆ V is P3-convex if every vertex of V\U has at most one neighbor in U. For any set W ⊆ V , we use σ(W) to denote the unique minimal P3-convex set of vertices that contains W. Two players play the connected P3-game on a graph by alternately selecting unmarked vertices. At the start of the game all vertices are unmarked, and the set of marked vertices, denoted by M, is empty. A move consists of marking a previously unmarked vertex v, so that M is updated to M0 = σ(M ∪{v}), and such a move is legal only when M0 is connected. When it is a player’s turn to move, he loses the game if all vertices are marked. In this thesis, we consider the connected P3-game on grids and tori, and give closed form formulae of their corresponding Grundy values. We also consider the related Pr-game, with r ≥ 3, that is played on a path or a cycle.

Contents
1 Introduction 1
1.1 Impartial Combinatorial Games 2
1.2 Nim Game 3
1.3 P3 Games 4

2 Preliminaries 8
2.1 P-positions and N-positions 8
2.2 The Sprague-Grundy Function 10

3 Connected P3-Game on Grids 14
3.1 When v is a Corner Vertex 21
3.2 When v Lies on the Border 23
3.3 When v Lies in the Middle 30
3.4 All in a Nutshell 35

4 Connected P3-Game on Tori 37

5 Connected Pr-Game on Paths and Cycles 41
5.1 Connected Pr-Game on Paths 43
5.1.1 Properties of ⊕ 44
5.1.2 Case Analysis of the Closed Form 45
5.2 Connected Pr-Game on Cycles 49

6 Conclusion 52

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