Complete Forcing Numbers of Spiro Hexagonal Systems

Abstract

A forcing set of a perfect matching M of a graph G is a subset of M contained in no other perfect matchings of G. A global forcing set of G, introduced by Vukičević et al., is a subset of E(G) on which there are distinct restrictions of any two different perfect matchings of G. Combining the above“forcing” and “global” ideas, Xu et al. [Journal of Combinatorial Optimization 29, no. 4 (2015): 803–14] introduced a complete forcing set of G defined as a subset of E(G) on which the restriction of any perfect matching M of G is a forcing set of M. The minimum cardinality of complete forcing set is the complete forcing number of G. In this paper, we give the explicit expressions for the complete forcing numbers of several classes of spiro hexagonal systems. Moreover, we also discuss some propositions of spiro cata-condensed hexagonal system with perfect matching related to the concept of forcing numbers.

Keywords:

• Complete forcing number
• global forcing number
• spiro hexagonal systems

Acknowledgment

The authors thank the anonymous referees for their helpful suggestions to improve the exposition.

Additional information

Funding

This work was supported by NSFC (Grant Nos. 11761070, 61662079, and 11361062), Xinjiang Normal University Undergraduate teaching project (SDJG2017-3), The 13th Five-Year Plan for Key Discipline Mathematics (No. 17SDKD11), Xinjiang Normal University, and Outstanding Young Teachers Scientific Research Foundation of Xinjiang Normal University (XJNU201416).