Deniz, ZakirEkim, Tınaz2020-05-012020-05-0120190166-218X1872-6771https://doi.org/10.1016/j.dam.2018.09.033https://hdl.handle.net/20.500.12684/566810th International Colloquium on Graphs and Optimization (GO) -- JUL 10-14, 2016 -- Lucerne, SWITZERLANDDeniz, Zakir/0000-0002-0701-0397; Ekim, Tinaz/0000-0002-1171-9294WOS: 000468260200013A graph G is equimatchable if every maximal matching of G has the same cardinality. We are interested in equimatchable graphs such that the removal of any edge from the graph preserves the equimatchability. We call an equimatchable graph G edge-stable if G \ e, that is the graph obtained by the removal of edge e from G, is also equimatchable for any e is an element of E(G). After noticing that edge-stable equimatchable graphs are either 2-connected factor-critical or bipartite, we characterize edge-stable equimatchable graphs. This characterization yields an O(min(n(3.376), n(1.5)m)) time recognition algorithm. Lastly, we introduce and shortly discuss the related notions of edge-critical, vertex-stable and vertex critical equimatchable graphs. In particular, we emphasize the links between our work and the well-studied notion of shedding vertices, and point out some open questions. (C) 2018 Elsevier B.V. All rights reserved.en10.1016/j.dam.2018.09.033info:eu-repo/semantics/closedAccess1-well-coveredMaximal matchingEdge-stabilityEdge-criticalityShedding vertexArticle261136147WOS:000468260200013Q2Q3