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Independent mutual-visibility sets and distance edge-critical graphs

Jing Tian, Csilla Bujtás, Sandi Klavžar

Abstract

In this paper, connections between independent sets and the variety of mutual-visibility sets are studied. It is proved that every outer mutual-visibility set of a graph is independent if and only if the graph is distance edge-critical. Several constructions yielding distance edge-critical graphs are given. Graphs in which every independent set is a total mutual-visibility or a dual mutual-visibility set are characterized, as well as graphs in which every total mutual-visibility set is independent. Along the way the total mutual-visibility number of some graphs derived from fullerenes is determined. Graphs in which every independent set is a mutual-visibility set are discussed and characterized over diameter four graphs. It is proved that determining the maximum cardinality of an independent mutual-visibility set and deciding whether it equals the independence number of a graph are NP-hard problems, and the same is true for independent total, outer and dual mutual-visibility sets.

Independent mutual-visibility sets and distance edge-critical graphs

Abstract

In this paper, connections between independent sets and the variety of mutual-visibility sets are studied. It is proved that every outer mutual-visibility set of a graph is independent if and only if the graph is distance edge-critical. Several constructions yielding distance edge-critical graphs are given. Graphs in which every independent set is a total mutual-visibility or a dual mutual-visibility set are characterized, as well as graphs in which every total mutual-visibility set is independent. Along the way the total mutual-visibility number of some graphs derived from fullerenes is determined. Graphs in which every independent set is a mutual-visibility set are discussed and characterized over diameter four graphs. It is proved that determining the maximum cardinality of an independent mutual-visibility set and deciding whether it equals the independence number of a graph are NP-hard problems, and the same is true for independent total, outer and dual mutual-visibility sets.
Paper Structure (11 sections, 18 theorems, 5 equations, 4 figures)

This paper contains 11 sections, 18 theorems, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Outer mutual-visibility and independence
  4. Edge-critical graphs
  5. Total/dual mutual-visibility and independence
  6. Dense diamond graphs
  7. Mutual-visibility and independence
  8. Algorithmic time complexity of related problems
  9. Construction of $G_d(F)$.
  10. Graphs of diameter three.
  11. Open problems

Key Result

Proposition 1.1

cicerone-2024b If $G$ is a connected graph of order at least $3$ and with $g(G) \ge 5$, then every outer MV set of $G$ is independent.

Figures (4)

  • Figure 1: Wagner graph $M_8$
  • Figure 2: From left to right: $G_7 \cong H_{3,1}$; the graph obtained from $G_7$ by duplicating once each of its degree two vertices; $H_{4,2}$
  • Figure 3: An independent set of a diameter five graph which is not a MV set
  • Figure 4: Graph $G_5(F)$ for $F = (x_1 \lor x_2 \lor \overline{x}_3) \land (\overline{x}_1 \lor \overline{x}_2 \lor x_4) \land(x_2 \lor x_3 \lor \overline{x}_4)$. Black vertices form an independent total MV set of cardinality $\alpha(G_5(F))$.

Theorems & Definitions (21)

  • Proposition 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Theorem 4.1
  • Theorem 4.2
  • Proposition 4.3
  • ...and 11 more