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Local edge domination critical graphs. (English) Zbl 0870.05035

Discrete Math. 161, No. 1-3, 175-184 (1996); erratum ibid. 310, No. 10-11, 1648 (2010).
Summary: D. P. Sumner and P. Blitch [J. Comb. Theory, Ser. B 34, 65-76 (1983; Zbl 0512.05055)] defined a graph \(G\) to be \(k\)-\(\gamma\)-critical if \(\gamma(G)=k\) and \(\gamma(G+uv)=k-1\) for each pair \(u,v\) of nonadjacent vertices of \(G\). We define a graph to be \(k\)-\((\gamma,d)\)-critical if \(\gamma(G)=k\) and \(\gamma(G+uv)=k-1\) for each pair \(u,v\) of nonadjacent vertices of \(G\) that are at distance at most \(d\) apart. The 2-\((\gamma, 2)\)-critical graphs are characterized. Sharp upper bounds on the diameter of 3-\((\gamma,2)\)- and 4-\((\gamma,2)\)-critical graphs are established and partial characterizations of 3-\((\gamma,2)\)-critical graphs are obtained.

MSC:

05C35 Extremal problems in graph theory
05C75 Structural characterization of families of graphs

Citations:

Zbl 0512.05055
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References:

[1] Blitch, P., Domination in graphs, (Dissertation (1983), University of South Carolina) · Zbl 0512.05055
[2] Chartrand, G.; Lesniak, L., Graphs and Digraphs (1986), Wadsworth and Brooks/Cole: Wadsworth and Brooks/Cole Monterey, CA · Zbl 0666.05001
[3] Favaron, O.; Sumner, D. P.; Wojcicka, E., The diameter of domination \(k\)-critical graphs, J. Graph Theory, 18, 723-734 (1994) · Zbl 0807.05042
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[5] Sumner, D. P.; Blitch, P., Domination critical graphs, J. Combin. Theory B, 34, 65-76 (1983) · Zbl 0512.05055
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