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Relations between global forcing number and maximum anti-forcing number of a graph

Published: 15 April 2022 Publication History

Abstract

The global forcing number of a graph G is the minimal cardinality of an edge subset discriminating all perfect matchings of G, denoted by g f ( G ). For a perfect matching M of G, the minimal cardinality of an edge subset S ⊆ E ( G ) ∖ M such that G − S has a unique perfect matching is called the anti-forcing number of M. The maximum anti-forcing number of G among all perfect matchings is denoted by A f ( G ). It is known that the maximum anti-forcing number of a hexagonal system equals the famous Fries number.
For a bipartite graph G, we show that g f ( G ) ≥ A f ( G ). Next we extend such result to Birkhoff–von Neumann graphs, whose perfect matching polytopes are characterized solely by nonnegativity and degree constraints, by revealing an odd dumbbell of non-bipartite graphs with a unique perfect matching and minimum degree at least two. Finally, we obtain tight upper and lower bounds on g f ( G ) − A f ( G ). For a connected bipartite graph G with 2 n vertices, 0 ≤ g f ( G ) − A f ( G ) ≤ 1 2 ( n − 1 ) ( n − 2 ). For non-bipartite case, we have − O c c ( G ) ≤ g f ( G ) − A f ( G ) ≤ ( n − 1 ) ( n − 2 ) by introducing a new nonnegative parameter O c c ( G ).

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  • (2023)Complete forcing numbers of complete and almost-complete multipartite graphsJournal of Combinatorial Optimization10.1007/s10878-023-01078-746:2Online publication date: 24-Aug-2023

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        Published In

        cover image Discrete Applied Mathematics
        Discrete Applied Mathematics  Volume 311, Issue C
        Apr 2022
        154 pages

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        Elsevier Science Publishers B. V.

        Netherlands

        Publication History

        Published: 15 April 2022

        Author Tags

        1. Perfect matching
        2. Perfect matching polytope
        3. Birkhoff–von Neumann graph
        4. Solid brick
        5. Maximum anti-forcing number
        6. Global forcing number

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        • (2023)Complete forcing numbers of complete and almost-complete multipartite graphsJournal of Combinatorial Optimization10.1007/s10878-023-01078-746:2Online publication date: 24-Aug-2023

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