The Anti-forcing Number Of Cyclo-hexagonal Chains

Let G be a simple, connected, finite graph, M an edge-subset of G. A subset M is called a perfect matching(or Kekulé structure), if M covers all the vertices of G and any two edges of M have no common vertices. Let S be a subset of E( G), if G-S has a unique perfect matching, then we call S an anti-forcing set of G. The smallest cardinality among all anti-forcing sets of G is the anti-forcing number of G, denoted by ? ?Gaf.In this paper, we mainly discuss the anti-forcing number of cyclo-hexagonal chains according to the parity of the number of segment. We show that the anti-forcing number of cyclo-hexagonal chains which has an even number segments is 2. For a cyclo-hexagonal chain with one segment, we prove that the anti-forcing number is 1 if it is not linear, and 2 if it is linear. For a cyclo-hexagonal chain with three segments, we obtain that the anti-forcing number is 1 if it has a segment with length 2, otherwise 2. Furthermore, we determine that the anti-forcing number of cyclo-fibonacene chains with n( n is odd) hexagons is ?n/ 3 ?? ?.