Anti-forcing Edge And Forcing Polynomial Of Coronoid Hexagonal Systems

A hexagonal system is a finite 2-connected plane graph in which every interior face is bounded by a regular hexagon of side length one.A Benzenoid system is a hexagonal system of which each interior face is a hexagon.A connected subgraph C of a hexagonal system is called a coronoid system if C has at least one non-hexagonal interior face(called a corona hole)and each edge is contained in a hexagon of C.A coronoid system with exactly one corona hole is a single coronoid system.A coronoid system with more than one corona hole is called a multiple coronoid system.Benzenoid systems and coronoid systems are widely used in the study of benzenoid hydrocarbons and coronoid hydrocarbons.A set of independent edges of a graph G is called a matching of G.A matching M of G is called a perfect matching if every vertex of G is incident with exactly one edge in M.A forcing set S of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G.The minimum cardinality over all forcing sets of M is called the forcing number of M,denoted by f(G,M).Opposite to the forcing number,Vukicevic and Trinajstic introduced the anti-forcing number.An anti-forcing set S of G is a set of edges of G such that the graph obtained from G by deleting all edges in S has a unique perfect matching.So an edge of G is an anti-forcing edge if itself forms an anti-forcing set.The smallest cardinality of anti-forcing sets of G is called the anti-forcing number,denoted by af(G).A hexagonal system with an anti-forcing edge if and only if it is a truncated parallelogram.In this paper,we study a kind of multiple coronoid systems with an anti-forcing edge.Let G be a catacondensed multiple coronoid system,then af(G)= 1 if and only if G can be obtained from a L-type hexagonal chain T by gluing more than one generalized catacondensed bezenoid system or coronoid system just along a horizontal cantilever and a slant cantilever of T.In 2015,Zhang etc.proposed the forcing polynomial of a graph G as a counting polynomial for perfect matching of G with the same forcing number.The number of perfect matchings of G equals the sum of all coefficients of the forcing polynomial.In this paper,we study the forcing polynomial of a kind of coronoid hexagonal chain and derive its recursive expression.The main contents of the paper can be divided into the following three parts:In the first section,we introduce the background of the topic,the current re-search status and the content of the research.In the second section,we study the multiple coronoid systems with one anti-forcing edge.In the third section,a recursive expressions for the forcing polynomial of a kind of single coronoid systems is given.