# On the $g$-good-neighbor connectivity of graphs

## TL;DR

This paper investigates the properties and bounds of the $g$-good-neighbor connectivity in graphs, characterizes specific cases, and establishes extremal results to enhance understanding of fault tolerance in interconnection networks.

## Contribution

It provides new bounds, characterizations, and extremal results for the $g$-good-neighbor connectivity, advancing the theoretical understanding of fault tolerance parameters.

## Key findings

- Bounds: $1 \,\leq\, \kappa^g(G) \leq n-2g-2$ for certain $g$.
- Characterizations of graphs with $\,\kappa^g(G)=1,2$ and trees with $\,\kappa^g(T_n)=n-t$.
- Extremal results for the $g$-good-neighbor connectivity.

## Abstract

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq \kappa^g(G)\leq n-2g-2$ for $0\leq g\leq \left\{\Delta(G),\left\lfloor \frac{n-3}{2}\right\rfloor\right\}$, and graphs with $\kappa^g(G)=1,2$ and trees with $\kappa^g(T_n)=n-t$ for $4\leq t\leq \frac{n+2}{2}$ are characterized, respectively. In the end, we get the three extremal results for the $g$-good-neighbor connectivity.