Structural Properties of Connected Domination Critical Graphs

A graph G is said to be k-γc-critical if the connected domination number γc(G) equal k and γc(G+uv)<k for any pair of non-adjacent vertices u v G. Let ζ cut let ζ0 maximum that can contained in one block. For an integer ℓ≥0, a ℓ-factor critical G−S has perfect matching subset S size ℓ. It was proved by Ananchuen 2007 k=3, Kaemawichanurat 2010 k=4 2020 k≥5 every at most k−2 graphs with were characterized. In 2020, further that, k≥4, satisfies inequality ζ0(G)≤mink+23,ζ. this paper, we characterize all having k−3 vertices. Further, establish realizability given 2≤ζ≤k−2 2≤ζ0≤mink+23,ζ, there exists block which contains Finally, odd order minimum degree two 1-factor only 1≤k≤2. K1,3-free even three 2-factor