A maximum resonant set of polyomino graphs

Zhang, Heping; Zhou, Xiangqian

doi:10.7151/dmgt.1857

Cited by 13 publications

(9 citation statements)

References 29 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: 54%

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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: 54%

“…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%

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%

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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“…For the maximum forcing number of Q n , Alon proved that for sufficiently large n this number is near to the total number of edges in a perfect matching of Q n (see Riddle (2002)), but its specific value is still unknown. Afterwards, Adams et al (2004) generalized Alon's result to a k-regular bipartite graph and for a hexagonal system, a polyomino graph or a (4, 6)fullerene, Xu et al (2013); Zhang and Zhou (2016); Shi et al (2017) showed that its maximum forcing number equals its Clar number, respectivey. For a graph G with a perfect matching, Lei et al (2016) connected the anti-forcing number and forcing number of a perfect matching of G, and showed that the maximum forcing number of G is no more than Af (G).…”

Section: Introductionmentioning

confidence: 96%

Tight upper bound on the maximum anti-forcing numbers of graphs

Shi,

Zhang

2017

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Let G be a simple graph with a perfect matching. Deng and Zhang showed that the maximum anti-forcing number of G is no more than the cyclomatic number. In this paper, we get a novel upper bound on the maximum anti-forcing number of G and investigate the extremal graphs. If G has a perfect matching M whose anti-forcing number attains this upper bound, then we say G is an extremal graph and M is a nice perfect matching. We obtain an equivalent condition for the nice perfect matchings of G and establish a one-to-one correspondence between the nice perfect matchings and the edge-involutions of G, which are the automorphisms α of order two such that v and α(v) are adjacent for every vertex v. We demonstrate that all extremal graphs can be constructed from K2 by implementing two expansion operations, and G is extremal if and only if one factor in a Cartesian decomposition of G is extremal. As examples, we have that all perfect matchings of the complete graph K2n and the complete bipartite graph Kn,n are nice. Also we show that the hypercube Qn, the folded hypercube F Qn (n ≥ 4) and the enhanced hypercube Q n,k (0 ≤ k ≤ n − 4) have exactly n, n + 1 and n + 1 nice perfect matchings respectively.

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A maximum resonant set of polyomino graphs

Cited by 13 publications

References 29 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)

Tight upper bound on the maximum anti-forcing numbers of graphs

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