Lit-Only σ-Game and its Dual Game on Tree

• Main Authors: Lin, Yu-Sheng, 林育生
• Other Authors: Weng, Chih-Wen
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/78175425391796314631

碩士 === 國立交通大學 === 應用數學系所 === 98 === Let G be a simple connected graph with n vertices {1, 2, … , n}. A configuration of G is an assignment of one of two colors, black or white, to each vertex of G. A move on the set of configurations of G is a function from the set to itself. Two different games with their own sets of moves are investigated in this thesis. The first one which is called the lit-only σ-game, contains n moves Li corresponding to the vertices i. When the move Li is applied to a configuration u, the color of a vertex j in u is changed if and only if i is a black vertex and j is a neighbor of i. The second one which is called the lit-only dual σ-game, has n moves Li* corresponding to the vertices i. When the move Li* is applied to a configuration u, the color of a vertex j in u is changed if and only if i has odd number of black neighbors and j=i. The dual relation between these two games will be clarified. In each of the two games, the set of configurations is partitioned into orbits by the action of its moves. An orbit with more than one configuration is called it nontrivial orbit. When G is a tree with some minor assumptions, we conjecture that there are two nontrivial lit-only dual σ-game orbits. We prove the conjecture under certain assumptions. It is known that the lit-only σ-game on a tree with perfect matchings has three orbits. We give an algorithm to describe these three orbits by applying the results in its dual game.