Anti-Forcing Spectra of Convex Hexagonal Systems

Abstract

For any perfect matching M of a graph AG, the anti-forcing number of M af(G,M) is the cardinality of a minimum edge subset S⊆E(G)\M such that the graph G−S has only one perfect matching. The anti-forcing numbers of all perfect matchings of G form its anti-forcing spectrum, denoted by Specaf(G). For a convex hexagonal system O(n1,n2,n3) with n1≤n2≤n3, denoted by H, it has the minimum anti-forcing number n1. In this paper, we derive a formula for its maximum anti-forcing number Af(H), i.e., the Fries number. Next, we prove that [n1,c]∪{c+2,c+4,…,Af(H)−2,Af(H)}⊆Specaf(H) for the specific integer c with the same parity as Af(H). In particular, we obtain that if n1+n2−n3≤1, then c=Af(H), which implies that Specaf(H)=[n1,Af(H)] is an integer interval. Finally, we also give some non-continuous situations: Specaf(O(2,n,n))=[2,4n−2]\{4n−3} for n≥2; the anti-forcing spectrum of H has a gap Af(H)−1 for n1=n2≥2 and n3 even, or n2=n3 and n1≥2 even.