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# Problems on matchings and independent sets of a graph

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. PDF Dis_Mat_341-6_1561_2018.pdf - Published Version Restricted to Registered users only Download (369kB) | Request a copy
Official URL: http://dx.doi.org/10.1016/j.disc.2018.02.021

## 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 Copy right for this article belong to ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 23 May 2018 14:55 23 May 2018 14:55 http://eprints.iisc.ac.in/id/eprint/59911 View Item