Journal of Computer Science and Technology

, Volume 15, Issue 5, pp 416–422

# Decision tree complexity of graph properties with dimension at most 5

• Gao Suixiang
• Lin Guohui
Article

## Abstract

A graph property is a set of graphs such that if the set contains some graphG then it also contains each isomorphic copy ofG (with the same vertex set). A graph propertyP onn vertices is said to be elusive, if every decision tree algorithm recognizingP must examine alln(n−1)/2, pairs of vertices in the worst case. Karp conjectured that every nontrivial monotone graph property is elusive. In this paper, this conjecture is proved for some cases. Especially, it is shown that if the abstract simplicial complex of a nontrivial monotone graph propertyP has dimension not exceeding 5, thenP is elusive.

## Keywords

monotone graph property decision tree complexity elusive

## References

1. [1]
Rosenberg A L. On the time required to recognize properties of graphs: A problem.SIGACT News, 1973, 5(1): 15–16.
2. [2]
Kahn J, Saks M, Strutevant D. A topological approach to evasiveness.Combinatorica, 1984, 4(2): 297–306.
3. [3]
Bollobas B. Complete subgraphs are elusive.J. Combinatorial Theory (Ser. B), 1976, 21(2): 1–7.
4. [4]
Kirkpatrick D. Determining graph properties from matrix representations. InProceedings of 6th SIGACT Conference, Seattle 1974,ACM, 1975, pp.84–90.Google Scholar
5. [5]
Milner E C, Welsh D J A. On the computational complexity of graph theoretical properties. InProceedings of 5th British Columbia Conf. Combinatorics, 1975, pp.471–487.Google Scholar
6. [6]
Triesch E. Some results on elusive graph properties.SIAM J. Computing, 1994, 23(1): 247–254.
7. [7]
8. [8]
Bollobas B, Eldridge S E. Problem. InProceedings of Fifth British Combinatorial Conf., Utilitas Math., Winnipeg, C. st. J. A. Nash-Williams, Sheehan J (eds.), 1976, pp. 689–691.Google Scholar