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Graph Configurations and Independent Bondage Numbers of Planar Graphs

E. G. K. M. Gamlath, Bing Wei, Talmage James Reid

Abstract

The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a graph with a larger independent domination number than that of G. In this research, we enhance the existing upper bound on the independent bondage number of a planar graph with a minimum degree of at least three by identifying specific configurations within such planar graphs.

Graph Configurations and Independent Bondage Numbers of Planar Graphs

Abstract

The independent domination number of a finite graph G is the minimum cardinality of an independent dominating set of vertices. The independent bondage number of G is the minimum cardinality of a set of edges whose deletion results in a graph with a larger independent domination number than that of G. In this research, we enhance the existing upper bound on the independent bondage number of a planar graph with a minimum degree of at least three by identifying specific configurations within such planar graphs.
Paper Structure (4 sections, 8 theorems, 4 equations, 2 figures)

This paper contains 4 sections, 8 theorems, 4 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. The Discharging Rules for Theorem \ref{['maintheorem']}
  3. Preliminaries to prove Theorem \ref{['IBNconfigurationproof']}
  4. Proof of Theorem \ref{['IBNconfigurationproof']}

Key Result

Theorem 1

If $G$ is a planar graph with $\delta(G) \geq 3$, then $b_i(G) \leq 8$.

Figures (2)

  • Figure 1: Some configurations in a planar graph with minimum degree three (dashed links are paths of length at least one)
  • Figure 2: $1,2,3$, and $k$-fans around $v\cup N(v)$ (dashed links are paths of length at least one)

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Proposition 1
  • proof
  • proof
  • proof
  • proof
  • ...and 17 more