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    Critical equimatchable graphs
    (University of Queensland, 2024) Deniz, Z.; Ekim, T.
    A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge creates a graph that is not equimatchable, called edge-critical equimatchable graphs (ECE-graphs). We show that apart from two simple cases, namely bipartite ECE-graphs and even cliques, all ECE-graphs are 2-connected factor-critical. Accordingly, we give a characterization of factor-critical ECE-graphs with connectivity 2. Our result provides a partial answer to an open question posed by Levit and Mandrescu [Eur. J. Comb. 20 (2019), 261–272] on the characterization of well-covered graphs with no shedding vertex. We also introduce equimatchable graphs such that the removal of any vertex creates a graph that is not equimatchable, called vertex-critical equimatchable graphs (VCEgraphs). To conclude, we clarify the relationship between various subclasses of equimatchable graphs (including ECE-graphs and VCE-graphs) and discuss the properties of factor-critical ECE-graphs with connectivity at least 3. © The author(s).
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    On Some Subclasses of Split B1 -EPG Graphs
    (Springer Science and Business Media Deutschland GmbH, 2020) Deniz, Z.; Nivelle, S.; Ries, B.; Schindl, D.
    In this paper, we are interested in edge intersection graphs of paths in a grid, such that each path has at most one bend. These graphs were introduced in[14] and they are called B1-EPG graphs. We focus on split B1-EPG graphs, and study subclasses defined by restricting the paths to subsets of the four possible shapes (?, ?, ? and ? ). We first state that the set of minimal forbidden induced subgraphs for the class of split ? -path graphs is infinite. Then, we further focus on two subclasses, and provide finite forbidden induced subgraphs characterizations for all possible subclasses defined by restricting to any subset of shapes. © 2020, Springer Nature Switzerland AG.