Regular Graphs with Maximum Forest Number

Let G be a graph and F ⊆ V (G). Then F is called an induced forest of G if 〈G〉 contains no cycle. The forest number, f(G), of G is defined by f(G) := max{|F | : F is an induced forest of G}. It was proved by the second author in [6] that if G is an r-regular graph of order n, then f(G) ≤ b nr−2 2(r−1)c. It was also proved that the bound is sharp by constructing an r-regular graph H of order n with f(H) = b nr−2 2(r−1)c. In this paper we consider the problem of determining which r-regular graphs G of order n have the forest number b nr−2 2(r−1)c. The problem was asked by Bau and Beineke [1] for r = 3 and, in this particular case, it was answered by the second author in [7]. We are able to answer the problem for all r ≥ 4. More precisely, we are able to obtain an algorithm of finding all r-regular graphs G of order n with f(G) = b nr−2 2(r−1)c. Furthermore, we prove that if R(r; f = b nr−2 2(r−1)c) is the set of all r-regular graphs G of order n with f(G) = b nr−2 2(r−1)c and G1, G2 ∈ R(r ; f = b nr−2 2(r−1)c), then there exists a sequence of switchings σ1, σ2, . . . , σt such that for each i = 1, 2, . . . , t, G12i 1 ∈ R(r; f = b nr−2 2(r−1)c) and G σ1σ2...σt 1 = G2.