Abstract
A matching M of a graph G is maximal if it is not a proper subset of any other matching in G. Maximal matchings are much less known and researched than their maximum and perfect counterparts. In this paper we present the recurrences and generating functions for the sequences enumerating maximal matchings in two classes of chemically interesting linear polymers: polyspiro chains and benzenoid chains. We also analyze the asymptotic behavior of those sequences and determine the extremal cases.
Highlights
A matching in a graph is a collection of its edges such that no two edges in this collection have a vertex in common
A strong initial impetus to their study came from the chemistry of benzenoid compounds after it was observed that the stability of benzenoid compounds is related to the existence and the number of perfect matchings in the corresponding graphs
We provide enumerative and extremal results on maximal matchings in two classes of linear polymers of chemical interest: the polyspiro chains and benzenoid chains
Summary
A matching in a graph is a collection of its edges such that no two edges in this collection have a vertex in common. Further motivation came from the statistical mechanics via the Kasteleyn’s solution of the dimer problem [15, 16] and its applications to evaluations of partition functions for a given value of temperature In both cases, the matchings under consideration are perfect, i.e., their edges are collectively incident to all vertices of G. See [11] for a survey of independent domination in graphs and [19] for perfect and efficient edge domination In spite of their obscurity, maximal matchings are natural models for several problems connected with adsorption of dimers on a structured substrate and block-allocation of a sequential resource. We provide enumerative and extremal results on maximal matchings in two classes of linear polymers of chemical interest: the polyspiro chains and benzenoid chains. We end by comparing our results with enumerative results for other types of structures in similar polymers and by discussing some possible directions of future research
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