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On Camby-Plein's Characterization of Domination Perfect Graphs

Vadim Zverovich

TL;DR

The paper critiques Camby and Plein's 2017 claims about counterexamples to the 1995 characterization of domination perfect graphs, showing their counterexamples are invalid and the proposed new characterization is incorrect. It reinforces the established criterion that a graph is domination perfect if $\gamma(H)=i(H)$ for all induced subgraphs $H$, and provides a concise proof of the 1995 result using a minimum counterexample and a partition-based argument, culminating in a contradiction if forbidden subgraphs $G_1$–$G_{17}$ are absent. The work also highlights a practical consequence: a polynomial-time method to convert any dominating set into an independent dominating set of the same or smaller size for domination-perfect graphs. Overall, it clarifies the landscape of domination-perfect graphs, reaffirming known theorems while exposing flaws in the proposed newer characterizations.

Abstract

We show that all results stated in [E. Camby, F. Plein, Discrete Appl. Math. 217 (2017) 711-717] are either previously known or incorrect. For example, Camby and Plein claimed to provide counterexamples to the 1995 characterization of domination perfect graphs due to Zverovich and Zverovich; however, these counterexamples are not valid. Moreover, the new characterization of domination perfect graphs proposed in that paper is incorrect. For completeness, we present a relatively brief proof of the 1995 characterization of domination perfect graphs due to Zverovich and Zverovich.

On Camby-Plein's Characterization of Domination Perfect Graphs

TL;DR

The paper critiques Camby and Plein's 2017 claims about counterexamples to the 1995 characterization of domination perfect graphs, showing their counterexamples are invalid and the proposed new characterization is incorrect. It reinforces the established criterion that a graph is domination perfect if for all induced subgraphs , and provides a concise proof of the 1995 result using a minimum counterexample and a partition-based argument, culminating in a contradiction if forbidden subgraphs – are absent. The work also highlights a practical consequence: a polynomial-time method to convert any dominating set into an independent dominating set of the same or smaller size for domination-perfect graphs. Overall, it clarifies the landscape of domination-perfect graphs, reaffirming known theorems while exposing flaws in the proposed newer characterizations.

Abstract

We show that all results stated in [E. Camby, F. Plein, Discrete Appl. Math. 217 (2017) 711-717] are either previously known or incorrect. For example, Camby and Plein claimed to provide counterexamples to the 1995 characterization of domination perfect graphs due to Zverovich and Zverovich; however, these counterexamples are not valid. Moreover, the new characterization of domination perfect graphs proposed in that paper is incorrect. For completeness, we present a relatively brief proof of the 1995 characterization of domination perfect graphs due to Zverovich and Zverovich.
Paper Structure (2 sections, 23 theorems, 5 equations, 7 figures)

This paper contains 2 sections, 23 theorems, 5 equations, 7 figures.

Table of Contents

  1. Summary of Results on Domination Perfect Graphs
  2. Proof of Theorem \ref{['ZZ']}

Key Result

Theorem 1

If $G$ has no induced subgraph isomorphic to $K_{1,3}$, then $\gamma (G) = i (G)$.

Figures (7)

  • Figure 1: Graph $S$.
  • Figure 2: Graphs $T_1$ and $T_2$.
  • Figure 3: Graphs $G_1-G_{17}$.
  • Figure 4: Graphs $H_0-H_9$Cam.
  • Figure 5: Graphs $U_1$ (left) and $U_2$ (right).
  • ...and 2 more figures

Theorems & Definitions (23)

  • Theorem 1: Allan and Laskar All
  • Theorem 2: Sumner and Moore Sum1Sum2
  • Theorem 3: Sumner and Moore Sum1Sum2
  • Theorem 4: Sumner and Moore Sum1Sum2
  • Theorem 5: Sumner and Moore Sum1Sum2
  • Theorem 6: Bollobás and Cockayne Bol
  • Theorem 7: Zverovich and Zverovich Zve0
  • Corollary 1: Zverovich and Zverovich Zve0
  • Theorem 8: Topp and Volkmann Top
  • Theorem 9: Zverovich and Zverovich Zve1
  • ...and 13 more