Geodesic
Edge Dominating Sets and Vertex Covers
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 437-456

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Bipartite graphs with equal edge domination number and maximum matching cardinality are characterized. These two parameters are used to develop bounds on the vertex cover and total vertex cover numbers of graphs and a resulting chain of vertex covering, edge domination, and matching parameters is explored. In addition, the total vertex cover number is compared to the total domination number of trees and grid graphs.
Keywords: edge dominating set, matching, total dominating set, vertex cover 
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     title = {Edge {Dominating} {Sets} and {Vertex} {Covers}},
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Dutton, Ronald; Klostermeyer, William F. Edge Dominating Sets and Vertex Covers. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 437-456. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a15/