The smallest nonevasive graph property

TL;DR

This paper identifies the smallest nontrivial nonevasive property of graphs, demonstrating it for 5-vertex graphs and establishing its uniqueness for graphs with at most 5 vertices.

Contribution

The authors construct a nontrivial nonevasive property for 5-vertex graphs and prove its uniqueness among properties with up to 5 vertices.

Findings

Existence of a nontrivial, nonevasive property for 5-vertex graphs

Uniqueness of this property for graphs with at most 5 vertices

Extension of understanding of evasiveness in graph properties

Abstract

A property of n-vertex graphs is called evasive if every algorithm testing this property by asking questions of the form "is there an edge between vertices u and v" requires, in the worst case, to ask about all pairs of vertices. Most "natural" graph properties are either evasive or conjectured to be such, and of the few examples of nontrivial nonevasive properties scattered in the literature the smallest one has n=6. We exhibit a nontrivial, nonevasive property of 5-vertex graphs and show that it is essentially the unique such with n at most 5.