Biclique-free graph

In graph theory, a branch of mathematics, a t-biclique-free graph is a graph that has no K_t,t (complete bipartite graph with 2t vertices) as a subgraph. A family of graphs is biclique-free if there exists a number t such that the graphs in the family are all t-biclique-free. The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity.

According to the Kővári–Sós–Turán theorem, every n-vertex t-biclique-free graph has O(n^{2 − 1/t}) edges, significantly fewer than a dense graph would have.^[1] Conversely, if a graph family is defined by forbidden subgraphs or closed under the operation of taking subgraphs, and does not include dense graphs of arbitrarily large size, it must be t-biclique-free for some t, for otherwise it would include large dense complete bipartite graphs.

As a lower bound, Erdős, Hajnal & Moon (1964) conjectured that every maximal t-biclique-free bipartite graph (one to which no more edges can be added without creating a t-biclique) has at least (t − 1)(n + m − t + 1) edges, where n and m are the numbers of vertices on each side of its bipartition.^[2]

A graph with degeneracy d is necessarily (d + 1)-biclique-free. Additionally, any nowhere dense family of graphs is biclique-free. More generally, if there exists an n-vertex graph that is not a 1-shallow minor of any graph in the family, then the family must be n-biclique-free, because all n-vertex graphs are 1-shallow minors of K_n,n. In this way, the biclique-free graph families unify two of the most general classes of sparse graphs.^[3]

In discrete geometry, many types of incidence graph are necessarily biclique-free. As a simple example, the graph of incidences between a finite set of points and lines in the Euclidean plane necessarily has no K_2,2 subgraph.^[4]

Biclique-free graphs have been used in parameterized complexity to develop algorithms that are efficient for sparse graphs with suitably small input parameter values. In particular, finding a dominating set of size k, on t-biclique-free graphs, is fixed-parameter tractable when parameterized by k + t, even though there is strong evidence that this is not possible using k alone as a parameter. Similar results are true for many variations of the dominating set problem.^[3] It is also possible to test whether one dominating set of size at most k can be converted to another one by a chain of vertex insertions and deletions, preserving the dominating property, with the same parameterization.^[5]