Hexagonal Systems With Forcing Number Two

Let M be a perfect matching of G. A subset S of M is called a forcing set of M if it does not belong to any other perfect matchings of G. The forcing number of M, denoted by f(G, M), is the number of edges contained in the minimum forcing set of M. Let M denote the set of all of the perfect matchings of G. The forcing number of G is defined as f(G)=min{f(G, M)|M∈M}. An edge e of G is called a forcing edge if it belongs to exactly one perfect matching of G. We call a hexagonal system with forcing edges a unit. P. Hansen, M. Zheng and F. Zhang, X. Li characterized the hexagonal systems with forcing edges independently in 1994 and 1995. In this paper, we first introduce a new definition-forcing region decomposition of the hexagonal system. Next, we obtain that for a hexagonal system H, f(H) = 2 if and only if H has a forcing region decomposition with two units. In the following, we obtain that any two units H₁ and H₂, if they are not one hexagon and straight hexagonal system, can be composed into one hexagonal system H with forcing number 2 such that H can be decomposed by H₁ and H₂ as two units. Finally, as an application and extension, we obtain the forcing numbers of some classes of hexagonal systems.