Theorem 3.7 Let G be a graph with c

Theorem 3.7 Let G be a graph with cycles, if all the cycles in G contains
a common non cutvertex u, then tc(G) = 2.
Proof : Let u be a common non cutvertex of the cycles in G, and let G1 = {[x, u] : [x, u] ∈ E(G)}, clearly G1 induces a tree isomorphic to K1,n, where n is the number of edges incident to u. Now let G2 = ⟨G u⟩, then u ∈/ V (G2) and since u is a non cutvertex G2 is connected. Now assume that G2 is not a tree, then G2 contains at least one cycle as a subgraph, let this cycle be C, since G2 is a subgraph of G then C is a subgraph of G, but all cycles in G contains u so u ∈ V(C), but V(C) ⊆ V(G2) so u ∈ V(G2) a contradiction because G2 = ⟨Gu⟩, thus G2 must be a tree. Hence the family FG = {G1,G2} is a tree cover of G. Thus tc(G) ≤ |FG| = 2, and since G is not a tree, tc(G) ≥ 2. Accordingly tc(G) = 2.