The Identification of Fundamental Subcycles in a Strongly Connected Tournament

Abstract

As more and more human-related variables such as preference, subjective decision making, sense of taste or smell, and so on, are involved in the discussion of measurement and evaluation, the theoretical exploration into the foundation for measurement from a broad perspective is becoming important.
One remarkable character of human-related variables can be seen in their binary relationhship. Almost always intransitivity appears. For example, from among a set of objects {A, B, C, …}, take a pair of them and ask the evaluator: “Which do you prefer?” (the pairwise comparison) The evaluator may prefer A to B and B to C among many such comparisons. There is often the case, however, when A and C are paired later casually, the evaluator chooses C rather than A! In such a case, the data are represented as an asymmetric directed graph known as “tournament” which has cycles
Here, we present an analysis of the strongly connected tournament, sometimes too complicated and difficult to analyse.
By the operation of “addition of point”, we clarify the structural relationship among strongly connected subtournaments, and show the fundamental properties on strongly connected tournament, which yieled some useful information on transitive subsets in the intransitive structure as a whole.