Bhattacharya, Amitava and Mondal, Anupam and Murthy, T Srinivasa
(2018)
*Problems on matchings and independent sets of a graph.*
In: DISCRETE MATHEMATICS, 341
(6).
pp. 1561-1572.

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## Abstract

Let G be a finite simple graph. For X subset of V(G), the difference of X, d(X) := vertical bar X vertical bar vertical bar N(X)vertical bar where N(X) is the neighborhood of X and max {d(X) : X subset of V(G)} is called the critical difference of G. X is called a critical set if d(X) equals the critical difference and ker(G) is the intersection of all critical sets. diadem(G) is the union of all critical independent sets. An independent set S is an inclusion minimal set with d(S) > 0 if no proper subset of S has positive difference. A graph G is called a Konig-Egervdry graph if the sum of its independence number alpha(G) and matching number mu(G) equals vertical bar V(G)vertical bar. In this paper, we prove a conjecture which states that for any graph the number of inclusion minimal independent set S with d(S) > 0 is at least the critical difference of the graph. We also give a new short proof of the inequality vertical bar ker(G)vertical bar + vertical bar diadem(G)vertical bar <= 2 alpha(G). A characterization of unicyclic non-Konig-Egervary graphs is also presented and a conjecture which states that for such a graph G, the critical difference equals alpha(G) mu(G), is proved. We also make an observation about ker(G) using Edmonds-Gallai Structure Theorem as a concluding remark. (C) 2018 Elsevier B.V. All rights reserved.

Item Type: | Journal Article |
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Additional Information: | Copy right for this article belong to ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |

Depositing User: | Id for Latest eprints |

Date Deposited: | 23 May 2018 14:55 |

Last Modified: | 23 May 2018 14:55 |

URI: | http://eprints.iisc.ac.in/id/eprint/59911 |

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