1-4. Cross IO Contradictions Happen

As in principle short-circuit graph method can be nothing else than what might be able to get a Hamiltonian tie by possible fortune, thorough examination in all numbers comes to be necessary after all. The difference between the ordinary method (a method in round robin) and short-circuit graph method is at such point that short-circuit graph method is able to run after a fortune, in other words it has an effective strategy to hasten the convergence as possible. If the three modes experiment is perfect, it comes that (1)if the graph is a Hamiltonian graph, it can obtain the answer rapidly by the short-circuit graph method, (2)if the graph is a non-Hamiltonian graph, it can be judged by the three modes experiment, but actually there are non-Hamiltonian graphs which can not be separated by any three kinds of experiment, and their range is broad.

When subgraph G' of graph G is a Hamiltonian tie of G, the following should hold.

Rule 1: G' is a spanning 2-factor subgraph of G.
Rule 2: G' has not elementary circuits more than 2.

Let us call these 2 rules Hamiltonian tie postulate (criterion). Furthermore, we will call an experiment to detect offenses completely against the rule 1 by perfect experiment and an experiment to detect offenses completely against the rules 1 and 2 by full experiment. k-Terminal division experiment, cut-set experiment and mesh-diagram experiment cannot detect offenses against the rule 2, while short-circuit graph method and series-parallel experiment are able to detect them. The experimental graph method (short-circuit graph method) is in this meaning (disregarding its costs) a full experiment. Besides, both of k-terminal division and cut-set experiment are perfect experiments if they are applied to every divisions of a graph.

We call k-terminal division experiment and cut-set experiment generically by IO experiment, and k-terminal division and cut-set division collectively by IO-division. Also, the basic experiment of short-circuit graph method is called factor experiment. Factor experiment is a full experiment and IO experiment is a perfect experiment. Factor experiment is a full experiment but since it cannot avoid deadlocks, it is not a complete experiment. That is, it can detect errors in determinative paths after the fact but cannot prevent before the fact. It is considered that the cause is the non-determinability of phenomena of deadlocks.

For the satisfaction of perfect experiment with IO experiments, it should be requested to make experiments for every k-terminal divisions and every cut-set divisions. However, since both of k-terminal division experiment and cut-set experiment cannot experiment every elements of the graph for reasons of economy, it is practically impossible to make perfect experiments. Moreover, even if a perfect experiment could be made, there might exist such cases that they could not avoid occurrences of IO contradictions. That is, they can detect errors in determinative paths after the fact but cannot prevent before the fact. This shows the presence of the non-determinability also in phenomena of IO contradictions. We call this non-deterministic causal obstacle cross-IO contradiction.

There is such a case that plural k-terminal divisions own a same terminal point jointly. We call such k-terminal divisions crossing-terminal divisions. Similarly, in the case of cut-set divisions there is a case that plural cut-sets own a branch jointly, and we call them crossing-cut-set divisions. Where there exists a crossing-IO-division, it is probable to occur cross-IO contradictions. With respect to cross-IO contradiction, there are both of an absolutely inevitable contradiction and a relative contradiction or an avoidable contradiction according to the selections. If a graph has an unavoidable cross-IO contradiction, the graph is a non-Hamiltonian graph. Up to the present, a judgment with non-retrial to detect absolute cross-IO contradictions has not been developed.