Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

We study graph properties which are testable for bounded degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is believed that almost all, even very simple graph properties require a large complexity to be tested for arbitrary (bounded degree) graphs. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. We call a graph family non-expanding if every graph in this family has a weak expansion (its expansion is O(1/ log n), where n is the graph size). A graph family is hereditary if it is closed under vertex removal. Similarly, a graph property is hereditary if it is closed under vertex removal. We call a graph property Π to be testable for a graph family F if for every graph G ∈ F , in time independent of the size of G we can distinguish between the case when G satis es property Π and when it is far from every graph satisfying property Π. In this paper we prove that in the bounded degree graph model, any hereditary property is testable if the input graph belongs to a hereditary and non-expanding family of graphs. As an application, our result implies that, for example, any hereditary property (e.g., k-colorability, H-freeness, etc.) is testable in the bounded degree graph model for planar graphs, graphs with bounded genus, interval graphs, etc. No such results have been known before, and prior to our work, very few graph properties have been known to be testable for general graph classes in the bounded degree graph model. ∗A preliminary version of this paper, entitled “On Testable Properties in Bounded Degree Graphs,” authored by the rst and third authors, appeared in the Proc. of 18 Symposium on Discrete Algorithm (SODA), New Orleans, Louisiana, 2007, 494-501. †Research supported in part by NSF ITR grant CCR-0313219, Centre for Discrete Mathematics and its Applications (DIMAP) and EPSRC grant EP/D063191/1, NSF DMS grant 0354600, and DFG grant Me 872/8-3. ‡Department of Computer Science, University of Warwick, Coventry, CV4 7AL, U.K. [email protected]. §Microsoft Research. asa [email protected]. ¶Department of Computer Science, University of Paderborn, 33102 Paderborn, Germany. [email protected]. Work done while the author was visiting Rutgers University.