The Independence Number Project: Α-bounds

A lower bound for the independence number of a graph is a graph invariant l such that, for every graph G, l(G) ≤ α(G). Similarly, an upper bound for the independence number is a graph invariant u such that, for every graph G, α(G) ≤ u(G). Many efficiently computable upper and lower bounds, called α-bounds here, have been published and these are surveyed in the following section. They can be used to predict the value of α. Suppose l1, l2, . . . , lk are efficiently computable lower bounds for the independence number of a graph; then l = max{l1, l2, . . . , lk} is also an efficiently computable lower bound for the independence number. Similarly, if u1, u2, . . . , um are efficiently computable upper bounds for the independence number, then u = min{u1, u2, . . . , um} is also an efficiently computable upper bound for the independence number. For some graphs G, l(G) = u(G) and, in such cases, it follows that the independence number α(G) = l(G) = u(G) can be directly computed from its bounds. For instance, consider the graph consisting of the cycle C4 with a diagonal. It is known that, for every graph αc ≤ α and, for every graph, α ≤ αf , where αc is the critical independence number and αf is the fractional independence number. These bounds are both efficiently computable and, for this graph, equal 2. Thus the theory implies that α = 2. New efficiently computable independence number bounds are also of practical interest: they can lead to faster independence number computations. New bounds can lead to new exact predictions of the independence number of a graph, without any need for computer search of subsets of vertices or calculating independence numbers of subgraphs of the given graph. If it is known that α must lie in the interval [l, u] then only subsets of sizes in this range must be considered. In some instances theoretical upper and lower bounds for α can be used to predict the independence number with no further search (in this case the theory predicts that α lies in an