On Maximum Induced Forests in Graphs

We consider the problem of determining the order of maximum induced forest, I(G), of a graph G. In this paper we prove that if R(d) is the graph of realizations of a degree sequence d and if G and G′ are adjacent in R(d), then |I(G) − I(G′)| ≤ 1. With the fact on the connectivity of the graph of realizations, it follows that for any graphic degree sequence d, there exist integers a and b such that d has a realization G with I(G) = c if and only if c is an integer satisfying a ≤ c ≤ b. Again, for given a graphic degree sequence d, we define min(I,d) and max(I,d) to be min(I,d) := min{I(G) : G ∈ R(d)} and max(I,d) := max{I(G) : G ∈ R(d)}. We are able to find formulae for max(I,d) for all regular degree sequences d.