Independence and Connectivity of Connected Domination Critical Graphs

TL;DR

This paper investigates properties of 3-connected domination critical graphs, establishing bounds on independence and connectivity, and exploring conditions under which these parameters are equal, with implications for Hamiltonian connectivity.

Contribution

It provides new bounds relating independence, connectivity, and minimum degree in 3-$\gamma_{c}$-critical graphs, and characterizes when these parameters are equal.

Findings

$ ext{Independence number } \alpha ext{ is at most } ext{connectivity } \kappa + 2$.

If $ ext{connectivity } \kappa ext{ is at least 3, then } ext{independence } ext{and minimum degree } ext{ are related by } ext{specific equalities}.

The bounds on $ ext{independence } ext{ relative to } ext{connectivity } ext{ are tight, leading to an open problem on Hamiltonian connectedness.

Abstract

A graph $G$ is said to be $k$-$\gamma_{c}$-critical if the connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Let $\delta, \kappa$ and $\alpha$ be respectively the minimum degree, the connectivity and the independence number. In this paper, we show that a $3$-$\gamma_{c}$-critical graph $G$ satisfies $\alpha \leq \kappa + 2$. Moreover, if $\kappa \geq 3$, then $\alpha = \kappa + p$ if and only if $\alpha = \delta + p$ for all $p \in \{1, 2\}$. We show that the condition $\kappa + 1 \leq \alpha \leq \kappa + 2$ is best possible to prove that $\kappa = \delta$. By these result, we conclude our paper with an open problem on Hamiltonian connected of $3$-$\gamma_{c}$-critical graphs.