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On graphs with unique geoodesics and antipodes

Dmitriy Gorovoy, David Zmiaikou

Abstract

In 1962, Oystein Ore asked in which graphs there is exactly one geodesic between any two vertices. He called such graphs geodetic. In this paper, we systematically study properties of geodetic graphs, and also consider antipodal graphs, in which each vertex has exactly one antipode (a farthest vertex). We find necessary and sufficient conditions for a graph to be geodetic or antipodal, obtain results related to algorithmic construction, and find interesting families of Hamiltonian geodetic graphs. By introducing and describing the maximal hereditary subclasses and the minimal hereditary superclasses of the geodetic and antipodal graphs, we get close to the goal of our research -- a constructive classification of these graphs.

On graphs with unique geoodesics and antipodes

Abstract

In 1962, Oystein Ore asked in which graphs there is exactly one geodesic between any two vertices. He called such graphs geodetic. In this paper, we systematically study properties of geodetic graphs, and also consider antipodal graphs, in which each vertex has exactly one antipode (a farthest vertex). We find necessary and sufficient conditions for a graph to be geodetic or antipodal, obtain results related to algorithmic construction, and find interesting families of Hamiltonian geodetic graphs. By introducing and describing the maximal hereditary subclasses and the minimal hereditary superclasses of the geodetic and antipodal graphs, we get close to the goal of our research -- a constructive classification of these graphs.
Paper Structure (12 sections, 35 theorems, 17 figures)

This paper contains 12 sections, 35 theorems, 17 figures.

Table of Contents

  1. Geodetic and antipodal graph check and bearing trees
  2. Geodetic graphs
  3. Blocks of geodetic graphs
  4. Bearing trees and stems
  5. Hamiltonian graphs
  6. Subclasses of Hamiltonian graphs
  7. Hereditary classes for geodetic graphs
  8. Antipodal graphs
  9. Weighted graphs
  10. Problems
  11. Isomorphic graphs
  12. Open problems

Key Result

lemma 1

Let $G$ be a connected graph on $n$ vertices. Then it is possible to check whether the graph $G$ is geodetic/antipodal or not for $O(n^3)$.

Figures (17)

  • Figure 1: Examples of geodetic graphs.
  • Figure 2: Examples of antipodal graphs.
  • Figure 3: A bearing tree with three balks.
  • Figure 4: A tree with five stems.
  • Figure 5: Two balks.
  • ...and 12 more figures

Theorems & Definitions (80)

  • definition 1
  • definition 2
  • definition 3
  • lemma 1
  • proof
  • definition 4
  • definition 5
  • lemma 2
  • proof
  • definition 6
  • ...and 70 more