グラフにおけるminimax, maximin, maximum, minimum点 : 廃棄物処理施設の建設地点に関する考察

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In this paper, I proposed that when we define an augmented graph, minimax, maximin, maximum, and minimum points can be found in the graph and the dual augmented graph can be defined. The "augmented graph" is a graph obtained by addition of some vertices and edges to a given graph which is called "basic graph" in this paper. If the vertices in a basic graph are denoted by v_i (i=1, 2, …, m), those added to the basic graph by W_j (j=1, 2, …, n), and the length of the edge between vertices w_j and v_i by d_<ji>, the minimax, maximin, maximum, and minimum points are the points w_j's which have the lengths d_<ji>^*, d_<ji>^<**>, S^*, and S^<**> defined by d_<jt>^* = <min>___j <max>___i d_<jt> d_<ji>^<**> = <max>___j <min>___i d_<ji> S^* = <max>___j Σ^m__<i=1>d_<ji> S^<**> = <min>___j Σ^m__<i=1>d_<ji> respectively. And the dual augmented graph is the graph obtained by regarding the w_j's (j=1, 2, …, n) as the vertices in a new basic graph and the v_i's (i=1, 2, …, m) as the vertices added to the new basic graph. The minimax, maximin, maximum, and minimum points are the points v_i's which have the lengths d_<ij>^*, d_<ij>^<**>, S^*, and S^<**> denned by d_<ij>^* = <min>___i <max>___j d_<ij> d_<ij>^<**> = <max>___i <min>___j d_<ij> S^* = <max>___i Σ^n__<j=1> d_<ij> S^<**> = <min>___i Σ^n__<j=1> d_<ij> respectively. Incidentally, using augmented and the dual augmented graphs, I tried to examine the positions of the facilities for disposal of garbage in Tokyo.
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グラフにおけるminimax, maximin, maximum, minimum点 : 廃棄物処理施設の建設地点に関する考察

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