Critical graphs with respect to total domination and connected domination
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A graph G is said to be k-γt -critical if the total domination number γt(G)= k and γt (G + uv) < k for every uv /∈ E(G). A k-γc-critical graph G is a graph with the connected domination number γc(G) = k and γc(G + uv) < k for every uv /∈ E(G). Further, a k-tvc graph is a graph with γt(G) = k and γt (G − v) < k for all v ∈ V(G), where v is not a support vertex (i.e. all neighbors of v have degree greater than one). A 2-connected graph G is said to be k-cvc if γc(G) = k and γc (G − v) < k for all v ∈ V(G). In this paper, we prove that connected k-γt -critical graphs and k-γc-critical graphs are the same if and only if 3 ≤ k ≤ 4. For k ≥ 5, we concentrate on the class of connected k-γt-critical graphs G with γc (G) = k and the class of k-γc-critical graphs G with γt(G) = k. We show that these classes intersect but they do not need to be the same. Further, we prove that 2-connected k-tvc graphs and k-cvc graphs are the same if and only if 3 ≤ k ≤ 4. Similarly, for k ≥ 5, we focus on the class of 2-connected k-tvc graphs G with γc (G) = k and the class of 2-connected k-cvc graphs G with γt (G) = k. We finish this paper by showing that these classes do not need to be the same.
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