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Corrections to classical results on Independence and Covering numbers of the Splitting graph

J. Castro, J. Leaños, O. Rosario

Abstract

The splitting graph $S(G)$ of a finite simple graph $G$ was introduced by Sampathkumar and Walikar in 1980~\cite{SW1980} and has been extensively studied in relation to graph invariants of $G$. In their original work, several formulas relating the independence number and the vertex cover number of $S(G)$ to the corresponding parameters of $G$ were stated and subsequently cited in the literature. In this paper, we show that some of these classical equalities do not hold in general. We present explicit counterexamples disproving the published results concerning independence and vertex cover numbers of splitting graphs. Moreover, we establish the correct formulas and precisely characterize the cases in which the original statements are valid and those in which they fail. These results correct an error that has remained unnoticed for more than four decades and provide a clearer understanding of splitting graphs from the perspective of independence and vertex cover number.

Corrections to classical results on Independence and Covering numbers of the Splitting graph

Abstract

The splitting graph of a finite simple graph was introduced by Sampathkumar and Walikar in 1980~\cite{SW1980} and has been extensively studied in relation to graph invariants of . In their original work, several formulas relating the independence number and the vertex cover number of to the corresponding parameters of were stated and subsequently cited in the literature. In this paper, we show that some of these classical equalities do not hold in general. We present explicit counterexamples disproving the published results concerning independence and vertex cover numbers of splitting graphs. Moreover, we establish the correct formulas and precisely characterize the cases in which the original statements are valid and those in which they fail. These results correct an error that has remained unnoticed for more than four decades and provide a clearer understanding of splitting graphs from the perspective of independence and vertex cover number.
Paper Structure (3 sections, 7 theorems, 2 equations)

This paper contains 3 sections, 7 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['t:main']}
  3. Counterexamples

Key Result

Proposition 1.1

(Proposition 3 in SW1980) For any graph $G$ with $n$ vertices, (i) $\alpha_0(S(G))=n=\beta_0(S(G))$. (ii) $\alpha_1(S(G))=2\alpha_1(G)$ and $\beta_1(S(G))=2\beta_1(G)$.

Theorems & Definitions (9)

  • Proposition 1.1
  • Definition 1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 3.1
  • proof