# Traceability of Connected Domination Critical Graphs

## TL;DR

This paper characterizes when connected domination critical graphs with a given number of cut-vertices have Hamiltonian paths, establishing a precise relationship between the criticality parameter, cut-vertices, and Hamiltonicity.

## Contribution

It provides a complete characterization of the existence of Hamiltonian paths in $k$-$rac{\gamma_c}{G}$-critical graphs based on their number of cut-vertices.

## Key findings

- Graphs with $rac{\gamma_c}{G}$-criticality and $rac{\zeta}{G}$ cut-vertices have Hamiltonian paths if and only if $rac{\zeta}{G}$ is between $k-3$ and $k-2$.
- The maximum number of cut-vertices in such graphs is $k-2$.
- The paper establishes a necessary and sufficient condition linking criticality, cut-vertices, and Hamiltonian paths.

## Abstract

A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex outside $S$ is adjacent to a vertex in $S$. A connected dominating set in $G$ is a dominating set $S$ such that the subgraph $G[S]$ induced by $S$ is connected. The connected domination number of $G$, $\gamma_c(G)$, is the minimum cardinality of a connected dominating set of $G$. 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 every pair of non-adjacent vertices $u$ and $v$ of $G$. Let $\zeta$ be the number of cut-vertices of $G$. It is known that if $G$ is a $k$-$\gamma_{c}$-critical graph, then $G$ has at most $k - 2$ cut-vertices, that is $\zeta \le k - 2$. In this paper, for $k \ge 4$ and $0 \le \zeta \le k - 2$, we show that every $k$-$\gamma_{c}$-critical graph with $\zeta$ cut-vertices has a hamiltonian path if and only if $k - 3 \le \zeta \le k - 2$.