Structural Properties of Connected Domination Critical Graphs

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

Clique doubly connected domination in the join and lexicographic product of graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

Nordhaus$-$Gaddum type results for connected and total domination

A dominating set of $G=(V,E)$ is a subset $S$ of $V$ such that every vertex in $V-S$ has at least one neighbor in $S.$ A connected dominating set of $G$ is a dominating set whose induced subgraph is connected. The minimum cardinality of a connected dominating set is the connected domination number $\gamma _{c}(G)$. Let $\delta ^{\ast }(G)=\min \{\delta (G),\delta (% \overline{G})\}$, where $\overline{G}$ is the complement of $G$ and $\delta (G)$ is the minimum vertex degree. In this paper, we improve upon existing results by providing new Nordhaus-Gaddum type results for connected domination. In particular, we show that if $G$ and $\overline{G}$ are both connected and $\min \{\gamma _{c}(G),\gamma _{c}(\overline{G})\}\geq 3$, then $\gamma _{c}(G)+\gamma _{c}(\overline{G})\leq 4+(\delta ^{\ast }(G)-1)(% \frac{1}{\gamma _{c}(G)-2}+\frac{1}{\gamma _{c}(\overline{G})-2})$ and $% \gamma _{c}(G)\gamma _{c}(\overline{G})\leq 2(\delta ^{\ast }(G)-1)(\frac{1}{% \gamma _{c}(G)-2}+\frac{1}{\gamma _{c}(\overline{G})-2}+\frac{1}{2})+4$. Moreover, we will establish accordingly results for total domination.