$k$-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$

TL;DR

This paper characterizes the minimal-edge graphs that remain resilient to vertex removal, ensuring the presence of multiple disjoint complete subgraphs, and explores their properties for various fault tolerance levels.

Contribution

It provides a characterization of the graphs with minimal edges that are $k$-vertex-fault-tolerant for $p$ disjoint complete graphs of order $c$, especially for $k=1$.

Findings

Identified minimal-edge $k$-fault-tolerant graphs for $k=1$ and $c \,\geq\, 3$.

Analyzed properties of such graphs for arbitrary $k$.

Established conditions for fault tolerance in disjoint complete graph structures.

Abstract

Vertex-fault-tolerance was introduced by Hayes~\cite{Hayes1976} in 1976, and since then it has been systematically studied in different aspects. In this paper we study $k$-vertex-fault-tolerant graphs for $p$ disjoint complete graphs of order $c$, i.e., graphs in which removing any $k$ vertices leaves a graph that has $p$ disjoint complete graphs of order $c$ as a subgraph. The main contribution is to describe such graphs that have the smallest possible number of edges for $k=1$, $p \geq 1$, and $c \geq 3$. Moreover, we analyze some properties of such graphs for any value of $k$.