Abstract
A graph G is said to be k-gamma(c)-critical if the connected domination number gamma(c)(G) is equal to k and gamma(c)(G+uv) < k for any pair of non-adjacent verticesuand v of G. Let xi be the number of cut vertices of Gand let xi(0) be the maximum number of cut vertices that can be contained in one block. For an integer l >= 0, a graph G is l-factor critical if G-Shas a perfect matching for any subset S of vertices of size l. It was proved by Ananchuen in 2007 for k = 3, Kaemawichanurat and Ananchuen in 2010 for k = 4 and by Kaemawichanurat and Ananchuen in 2020 for k = 5 that every k-gamma(c)-critical graph has at most k-2 cut vertices and the graphs with maximum number of cut vertices were characterized. In2020, Kaemawichanurat and Ananchuen proved further that, for k = 4, every k-gamma(c)-critical graphs satisfies the inequality xi(0)(G) <= min {[k + 2/3] , xi}. In this paper, we characterize all k-gamma(c)-critical graphs havingk-3 cut vertices. Further, we establish realizability that, for given k >= 4, 2 <= xi <= k-2 and 2 <= xi(0) <= min {[ k+ 2/3], xi}, there exists a k-gamma(c)-critical graph with?cut vertices having a block which contains ?(0) cut vertices. Finally, we proved that every k-gamma(c)-critical graph of odd order with minimum degree two is 1-factor critical if and only if 1 = k = 2. Further, we proved that every k-gamma(c)-critical K-1,K-3-free graph of even order with minimum degree three is 2-factor critical if and only if 1 <= k <= 2.