[Paper Review] Multiple vertex coverings by specified induced subgraphs

Zoltán Füredi, Dhruv Mubayi+1 more|ArXiv.org|Dec 22, 1999

Limits and Structures in Graph Theory5 references3 citations

TL;DR

This paper investigates the minimum order of a graph G that admits multiple induced subgraphs H₁,…,Hₖ covering all vertices, where each Hᵢ is an induced copy in G. It establishes a general upper bound of twice the sum of (|Hᵢ|−1) across i, and determines exact values for specific cases: when one Hᵢ is an independent set and the other is a star, the bound is tight and precisely characterized.

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.

Motivation & Objective

• To determine the minimum number of vertices in a graph G such that induced copies of specified graphs H₁,…,Hₖ collectively cover all vertices of G.
• To analyze the structural and extremal conditions under which such coverings are optimal.
• To establish tight bounds for specific graph pairs, particularly involving independent sets and stars.
• To generalize the vertex covering problem using induced subgraphs rather than arbitrary subgraphs.
• To explore the interplay between graph composition and induced subgraph coverage efficiency.

Proposed method

• Formalizing the problem as a minimum vertex cover problem using induced copies of specified graphs Hᵢ.
• Applying extremal combinatorics techniques to bound the minimum order of G in terms of the sizes of Hᵢ.
• Using double counting and induction to derive a general upper bound of 2∑(|Hᵢ|−1).
• Analyzing special cases via structural decomposition, particularly when one Hᵢ is an independent set.
• Employing constructive graph constructions to achieve tight bounds in the case of a star and an independent set.
• Leveraging known results in graph theory and extremal set systems to validate bounds and constructions.

Experimental results

Research questions

• RQ1 What is the minimum number of vertices in a graph G such that induced copies of k specified graphs H₁,…,Hₖ cover all vertices of G?
• RQ2 How does the minimum order of G scale with the sizes of the Hᵢ when k=2 and one Hᵢ is an independent set?
• RQ3 Can exact bounds be determined when the two graphs are a star and an independent set?
• RQ4 What is the tightness of the general upper bound of 2∑(|Hᵢ|−1) across different graph families?
• RQ5 How do structural properties of Hᵢ (e.g., connectivity, independence) affect the minimal covering graph size?

Key findings

• A general upper bound of 2∑(|Hᵢ|−1) is established for the minimum order of a graph G that induces coverings of V(G) by specified subgraphs Hᵢ.
• When k=2 and one Hᵢ is an independent set of size n, the minimum order of G is determined within a constant factor of 2n.
• For the case where one Hᵢ is a star and the other is an independent set, the exact minimum order of G is determined and shown to match the upper bound.
• The bound is tight in the star-independent set case, indicating that the construction is optimal.
• The results demonstrate that induced subgraph coverings can be efficiently constructed with minimal vertex count under specific structural constraints.
• The analysis reveals that the choice of Hᵢ significantly influences the minimal G, with independence and star structures yielding exact solutions.