Geodesic
(C3, C4, C5, C7)-Free Almost Well-Dominated Graphs
Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1099-1117

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The domination gap of a graph G is defined as the di erence between the maximum and minimum cardinalities of a minimal dominating set in G. The term well-dominated graphs referring to the graphs with domination gap zero, was first introduced by Finbow et al. [Well-dominated graphs: A collection of well-covered ones, Ars Combin. 25 (1988) 5–10]. In this paper, we focus on the graphs with domination gap one which we term almost well-dominated graphs. While the results by Finbow et al. have implications for almost well-dominated graphs with girth at least 8, we extend these results to (C3, C4, C5, C7)-free almost well-dominated graphs by giving a complete structural characterization for such graphs.
Keywords: well-dominated graphs, almost well-dominated graphs, domination gap 
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Alizadeh, Hadi; Gözüpek, Didem; Ekinci, Gülnaz Boruzanlı. (C3, C4, C5, C7)-Free Almost Well-Dominated Graphs. Discussiones Mathematicae. Graph Theory, Tome 42 (2022) no. 4, pp. 1099-1117. http://geodesic.mathdoc.fr/item/DMGT_2022_42_4_a5/