A graph \textit{ G} is a tuple (\textit{ V}, \textit{ E}), where \textit{ V} is the vertex set, \textit{ E} is the edge set. A reduced graph is a graph of deleting non-Hamiltonian edges and smoothing out the redundant vertices of degree 2 on an edge except for leaving only one vertex of degree 2. A 2-common (\textit{ v}, \textit{ 0}) combination is a cycle set in which every pair of joint cycles \textit{ A} and \textit{ B} satisfies $|V(A)\cap V(B)|=2$ and $|E(A)\cap E(B)|=0$. In this paper, we investigate the cycle structure of 2-common (\textit{ v}, \textit{ 0}) combination in reduced graphs, and give the characterizations of their Hamiltoncity.
Categories: cs.DM math.CO