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Maximal Independent Sets in Planar Triangulations

P. Francis, Abraham M. Illickan, Lijo M. Jose, Deepak Rajendraprasad

Abstract

We show that every planar triangulation on $n$ vertices has a maximal independent set of size at most $n/3$. This affirms a conjecture by Botler, Fernandes and Gutiérrez [Electron.\ J.\ Comb., 2024], which in turn would follow if an open question of Goddard and Henning [Appl.\ Math.\ Comput., 2020] which asks if every planar triangulation has three disjoint maximal independent sets were answered in the affirmative. Since a maximal independent set is a special type of dominating set (independent dominating set), this is a structural strengthening of a major result by Matheson and Tarjan [Eur.\ J.\ Comb., 1996] that every triangulated disc has a dominating set of size at most $n/3$, but restricted to triangulations.

Maximal Independent Sets in Planar Triangulations

Abstract

We show that every planar triangulation on vertices has a maximal independent set of size at most . This affirms a conjecture by Botler, Fernandes and Gutiérrez [Electron.\ J.\ Comb., 2024], which in turn would follow if an open question of Goddard and Henning [Appl.\ Math.\ Comput., 2020] which asks if every planar triangulation has three disjoint maximal independent sets were answered in the affirmative. Since a maximal independent set is a special type of dominating set (independent dominating set), this is a structural strengthening of a major result by Matheson and Tarjan [Eur.\ J.\ Comb., 1996] that every triangulated disc has a dominating set of size at most , but restricted to triangulations.

Paper Structure

This paper contains 7 sections, 4 theorems, 3 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Proof of Theorem \ref{['MainTheorem']}
  4. Case 1 ($d(v_1) = d(v_2) = d(v_3)= 4$).
  5. Case 2 ($d(v_1) =d(v_2) =4$, $d(v_3)= 5$).
  6. Case 3 ($d(v_1) =4$, $d(v_2) = d(v_3)= 5$).
  7. Concluding Remarks

Key Result

Theorem 1

Every $n$-vertex triangulation has an independent dominating set of size at most $n/3$.

Figures (1)

  • Figure 1: Two of the forbidden structures in a smallest counterexample to Theorem \ref{['MainTheorem']}.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • proof
  • Lemma 6
  • proof
  • Claim 6.1
  • ...and 1 more