# Minimum Vertex Cover of 2 vertex disjoint odd cycles that have edges between them

Consider the graph $$G$$, which is comprised of 2 vertex disjoint odd cycles ($$C_1$$, $$C_2$$) where $$|C_1|$$ and $$|C_2| \geq 5$$. $$G$$ is sub-cubic and connected, with edges in between the cycles. Because $$G$$ is sub-cubic, each node's degree $$\leq 3$$ but $$\geq 2$$.

I am interested to know whether the minimum vertex cover of $$G$$ can be found in polynomial time? Furthermore, I am also interested to know whether we can:

1. Establish a tight upper bound on the MinVC for such a graph.
We know that for either cycle $$C$$, $$MinVC = \left\lfloor\frac{|C|+1}{2}\right\rfloor$$. Thus, $$MinVC(G) \geq \left\lfloor\frac{|C_1|+1}{2}\right\rfloor+\left\lfloor\frac{|C_2|+1}{2}\right\rfloor$$
A loose upper bound would be to take all vertices of one cycle and solve optimally for the other. Thus, $$MinVC(G) \leq \min(|C_1|,|C_2|) + \left\lfloor\frac{\max(|C_1|, |C_2|)+1}{2}\right\rfloor$$.