On the maximum forcing and anti-forcing numbers of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml3" display="inline" overflow="scroll" altimg="si3.gif"><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mo>,</mml:mo><mml:mn>6</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>-fullerenes

Shi, Lingjuan; Wang, Hongwei; Zhang, Heping

doi:10.1016/j.dam.2017.07.009

Cited by 22 publications

(21 citation statements)

References 24 publications

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“…For hexagonal systems, Xu et al [34] proved that the maximum forcing number is equal to its resonant number. For polyomino graphs [18,39] and BN-fullerene graphs [26], the same result also holds. For more researches on the minimum and maximum forcing numbers, see [10,16,28,29,31,33,[35][36][37][38].…”

Section: Introductionsupporting

confidence: 53%

“…For polyomino graphs [18,39] and BN-fullerene graphs [26], the same result also holds. For more researches on the minimum and maximum forcing numbers, see [10,16,28,29,31,33,[35][36][37][38].…”

Section: Introductionsupporting

confidence: 53%

“…For general 2-connected plane bipartite graphs, Abeledo and Atkinson [1] obtained that the resonant number can be computed in polynomial time. Hence the maximum forcing numbers of hexagonal systems [34], polyomino graphs [39] and BN-fullerene graphs [26] can be computed in polynomial time.…”

Section: Conflict Of Interestmentioning

confidence: 99%

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Some tight bounds on the minimum and maximum forcing numbers of graphs

Liu¹,

Zhang²

2021

Preprint

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Let G be a simple graph with 2n vertices and a perfect matching. The forcing number of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G. Let f (G) and F (G) denote the minimum and maximum forcing number of G among all perfect matchings, respectively.Hetyei obtained that the maximum number of edges of graphs G with a unique perfect matching is n 2 (see Lovász [20]). We know that G has a unique perfect matching if and only if f (G) = 0. Along this line, we generalize the classical result to all graphs G with f (G) = k for 0 ≤ k ≤ n − 1, and obtain that the number of edges is at mostand characterize the extremal graphs as well. Conversely, we get a non-trivial lower bound of f (G) in terms of the order and size. For bipartite graphs, we gain corresponding stronger results. Further, we obtain a new upper bound of F (G).Finally some open problems and conjectures are proposed.

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Section: Introductionsupporting

confidence: 53%

Section: Introductionsupporting

confidence: 53%

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Some tight bounds on the minimum and maximum forcing numbers of graphs

Liu¹,

Zhang²

2021

Preprint

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“…For a hexagonal system with a perfect matching, Xu et al [22] showed that the maximum forcing number is equal to the Clar number, which can measure the stability of benzenoid hydrocarbons. Also, some similar results can be found in polyomino graphs [25] and (4,6)-fullerene graphs [15]. Furthermore, the maximum forcing numbers of some product graphs have been studied, such as rectangle grids P m × P n [2], cylindrical grids P m × C n [2,9], and tori C 2m × C 2n [11].…”

Section: Introductionmentioning

confidence: 69%

“…What's more, the minimum forcing numbers of tori C 2m × C 2n [14] and toroidal polyhexes [20] were solved. Sharp lower bounds for the minimum forcing numbers of (3,6)-fullerene graphs [16], (4,6)-fullerene graphs [8] and (5,6)-fullerene graphs [17,23] were revealed. Moreover, the forcing spectrum of square grids P 2n × P 2n [2] was also obtained.…”

Section: Introductionmentioning

confidence: 99%

Matching forcing polynomial of generalized Petersen graph GP(n, 2)

Zhao

2021

Preprint

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Harary et al. and Klein and Randić proposed the forcing number of a perfect matching in mathematics and chemistry, respectively. In detail, the forcing number of a perfect matching M of a graph G is the smallest cardinality of subsets of M that are contained in no other perfect matchings of G. The author and cooperators defined the forcing polynomial of G as the count polynomial for perfect matchings with the same forcing number of G, from which the average forcing number, forcing spectrum, and the maximum and minimum forcing numbers of G can be obtained. Up to now, a few papers have been considered on matching forcing problem of nonplane non-bipartite graphs. In this paper, we investigate the forcing polynomials of generalized Petersen graphs GP (n, 2) for n = 5, 6, . . . , 15, which is a typical class of non-plane non-bipartite graph.

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Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

Liu

Zhang

2023

Acta. Math. Sin.-English Ser.

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On the maximum forcing and anti-forcing numbers of(4,6)-fullerenes

Cited by 22 publications

References 24 publications

Some tight bounds on the minimum and maximum forcing numbers of graphs

Some tight bounds on the minimum and maximum forcing numbers of graphs

Matching forcing polynomial of generalized Petersen graph GP(n, 2)

Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

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