Abstract
A set Ⓢ of vertices of a graph G represents G if each edge of G is incident with at least one vertex of Ⓢ. A graph G is said to be edge-critical if the minimal number of vertices necessary to represent G decreases if any edge of G is omitted. Plummer [5] has given a method to construct an infinite family of edge-critical graphs with connectivity number 2. We use this method to construct a more extensive class of edge-critical graphs with connectivity number 2 and show that all edge-critical graphs with this connectivity number (≠K2) can be constructed from “smaller” edge-critical graphs. Finally we give examples of edge-critical graphs not constructable from smaller ones by this method.
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Wessel, W. Kanten-kritische graphen mit der zusammenhangszahl 2. Manuscripta Math 2, 309–334 (1970). https://doi.org/10.1007/BF01719589
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DOI: https://doi.org/10.1007/BF01719589