On the forced matching numbers of bipartite graphs

Adams, Peter; Mahdian, Mohammad; Mahmoodian, E. S.

doi:10.1016/j.disc.2002.10.002

Cited by 57 publications

(63 citation statements)

References 7 publications

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“…Recently, the minimum forcing numbers of bipartite graphs [1,2] have been extensively studied, especially for the stop signs [18], square grid [20], torus and hypercube [1,14,24]. To determine whether a set is a minimum forcing set of a perfect matching of a bipartite graph with maximum degree 3 is NP-complete [1]. Wang et al [28] gave a linear-time algorithm for computing the minimum forcing number of a toroidal polyhex according to its geometric structure.…”

Section: Introductionmentioning

confidence: 99%

“…The benzenoid systems with minimum forcing number 1 had been investigated in [19,[29][30][31], as well as plane bipartite graphs [33]. Recently, the minimum forcing numbers of bipartite graphs [1,2] have been extensively studied, especially for the stop signs [18], square grid [20], torus and hypercube [1,14,24]. To determine whether a set is a minimum forcing set of a perfect matching of a bipartite graph with maximum degree 3 is NP-complete [1].…”

Section: Introductionmentioning

confidence: 99%

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Forcing matching numbers of fullerene graphs

Zhang

Shiu

2010

Discrete Applied Mathematics

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a b s t r a c tThe forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig's classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Forcing matching numbers of fullerene graphs

Zhang

Shiu

2010

Discrete Applied Mathematics

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“…[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] But it should be mentioned that not all Kekule structures of general graphs admit a 6-forcing set, and consequently do not admit a 6-freedom. Such clearly is the circumstance with any multi-Kekulean graph without any 6-cycles -e.g., the 4-cycle (butadiene).…”

Section: Admissability Of 6-forcing and E6-forcingmentioning

confidence: 99%

“…He encouraged collaboration on the topic, which indeed ensued: a degree of concurrence of opinions was achieved, some initial results were established (on forcing, and some other ideas), and a paper subsequently appeared. 6 Following this work, forcing for Kekule structures has continued to receive attention, in both the mathematical, [7][8][9][10][11][12][13][14][15][16][17][18] and the chemical [19][20][21][22][23][24][25][26][27] literature. And further, what has been termed an "anti-forcing" idea has also been proposed 17 & studied.…”

Section: Hystory and Motivationmentioning

confidence: 99%

Forcing, Freedom, & Uniqueness in Graph Theory & Chemistry

Klein

Rosenfeld

2014

Croat. Chem. Acta

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Abstract. Harary's & Randić's ideas of "forcing" & "freedom" involve subsets of double bonds of Kekule structure such as to be unique to that Kekule structure. Such forcing sets are argued to be greatly generalizable to deal with various other coverings, and thence forcing seems to be fundamental, and of notable potential utility. Various forcing invariants associated to (molecular) graphs ensue, with illustrative (chemical) examples and some mathematical consequences being provided. A complementary "uniqueness" idea is noted, and the general characteristic of "derivativity" of "forcing" is established (as is relevant for QSPR fittings). Different ways in which different sorts of forcings arise in chemistry are briefly indicated.(doi: 10.5562/cca2000)

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“…And then the forcing concept is applied in various graph parameters viz. geodetic sets, hull sets, matching's, Steiner sets and edge covering in [3,4,6,8,10 ] by several authors. In this paper we study the forcing concept in minimum edge-to-vertex geodetic set of a connected graph.…”

Section: Introductionmentioning

confidence: 99%

The Forcing Edge-to-Vertex Geodetic Number of a Graph

Sujitha¹,

John²,

Vijayan³

2015

Int. J. of Pure and Appl. Math.

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For a connected graph G = (V, E), a set S ⊆ E is called an edge-to-vertex geodetic set of G if every vertex of G is either incident with an edge of S or lies on a geodesic joining a pair of edges of S. The minimum cardinality of an edge-to-vertex geodetic set of G is g ev (G). Any edge-to-vertex geodetic set of cardinality g ev (G) is called an edge-to-vertex geodetic basis of G. A subset T ⊆ S is called a forcing subset for S if S is the unique minimum edge-to-vertex geodetic set containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of S, denoted by f ev (S), is the cardinality of a minimum forcing subset of S. The forcing edge-to-vertex geodetic number of G, denoted by f ev (G), is f ev (G) = min {f ev (S)}, where the minimum is taken over all minimum edgeto-vertex geodetic sets S in G. Some general properties satisfied by the concept forcing edge-to-vertex geodetic number is studied. The forcing edge-to-vertex geodetic number of certain classes of graphs are determined. It is shown that

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On the forced matching numbers of bipartite graphs

Cited by 57 publications

References 7 publications

Forcing matching numbers of fullerene graphs

Forcing matching numbers of fullerene graphs

Forcing, Freedom, & Uniqueness in Graph Theory & Chemistry

The Forcing Edge-to-Vertex Geodetic Number of a Graph

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