Multiple vertex coverings by specified induced subgraphs

TL;DR

This paper investigates the minimum size of graphs that can be covered by specified induced subgraphs, providing bounds and exact solutions for particular cases involving independent sets and stars.

Contribution

It introduces a general upper bound for the minimum order of such graphs and determines exact solutions for specific graph pairs.

Findings

Established a general upper bound of twice the sum of m_i values.

Determined the exact minimum order for the case with two graphs, one being an independent set.

Solved the problem exactly when the pair consists of a star and an independent set.

Abstract

Given graphs H_1,...,H_k, we study the minimum order of a graph G such that for each i, the induced copies of H_i in G cover V(G). We prove a general upper bound of twice the sum of the numbers m_i, where m_i is one less than the order of H_i. When k=2 and one graph is an independent set of size n, we determine the optimum within a constant. When k=2 and the graphs are a star and an independent set, we determine the answer exactly.