Is star complexity a proxy for information based complexity of graphs?

Abstract

Information-based complexity (IBC) is a well-defined complexity measure of any object given a description in a language and a classifier that identifies those descriptions with the object. Of course, the exact numerical value will vary according to the descriptive language and classifier, but under certain universality conditions (eg the classifier identifies programs of a universal Turning machine that halt and output the same value), asymptotically, the complexity measure is independent of the classifier up to a constant of O(1). The hypothesis being investigated in this work that any practical IBC measure will similarly be asymptotically equivalent to any other practical IBC measure. Standish presented an IBC measure for graphs ${\cal C}$ that encoded graphs by their links, and identifies graphs as those that are automorphic to each other. An interesting alternate graph measure is {\em star complexity}, which is defined as the number of union and intersection operations of basic stars that can generate the original graph. Whilst not an IBC itself, it can be related to an IBC (called ${\cal C}^*$) that is strongly correlated with star complexity. In this paper, 10 and 22 vertex graphs are constructed up to a star complexity of 8, and the ${\cal C}^*$ compared emprically with ${\cal C}$. Finally, an easily computable upper bound of star complexity is found to be strongly related to ${\cal C}$.