By Brian Randell
ALGOL 60 implementation: the interpretation and use of ALGOL 60 courses on a computer
Volume five of A.P.I.C. reports in info processing
Issue five of ALGOL 60 Implementation, Brian Randell
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Additional info for ALGOL 60 Implementation: The Translation and Use of ALGOL 60 Programs on a Computer
Vn}, n > 1 of vertices. Place a coin ci onto vi for every i = 1,... ,n such that ci Cj whenever i j for some 1 i,j n. Let us say that a vertex is free if either it is covered by a coin cn or there exists another vertex that is covered by the same type of coin. ) Suppose that we are allowed to change the coins according to the following conditions: (1) For every i, j = 1,... ,n, we can put a coin ci onto the vertex vj if we have one of the following properties: (la) vk contains a coin ci and (vk, vj) E; (Ib) vj contains a coin ci.
N) = FF ( 1 , . . , n) = F ( l , . . , n — 1, n — 1) = (n — 1 , 1 , . . , n — 2, n — 1). , n. Now, let us consider the following procedure. , n. , 4. Then duplicate the coin C2 and put one of its copies to the vertex 3 (leaving the other one on the vertex 2). Then we get the configuration (c 1 , c2, c 3 , . . , c n - 1 ). Shift the coins right cyclically n — 1 times reaching (c 2 , c 2 , c 3 , . . , c n - 1 , c 1 ). Now we can shift the first m coins right cyclically m — 1 times, which results in ( c 2 , .
Thus Kn is the complete graph with n vertices and Kk,l is the complete bipartite graph with k and l. vertices (in that order). 1 (Kuratowski planar graph theorem). A graph is planar if and only if it contains no subdivision of K5, the complete graph with five vertices, nor of the complete bipartite graph K3,3. THE GRAPH K5 AND THE GRAPH K3,3 A realization of on the plane, according to the conditions mentioned, is called a topologicalplanar graph and is denoted R( }. The connected portions (in the topology of the considered plane P) of P \R( ) are called faces.