Edge-stable equimatchable graphs
|Title||Edge-stable equimatchable graphs|
|Publication Type||Journal Article|
|Year of Publication||2019|
|Authors||Deniz, Z., and T. Ekim|
|Journal||Discrete Applied Mathematics|
|Keywords||1-well-covered, Edge-criticality, Edge-stability, Maximal matching, Shedding vertex|
A 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∈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(n3.376,n1.5m)) 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.