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Characterizations of graph classes via convex geometries: a survey

Mitre C. Dourado, Marisa Gutierrez, Fábio Protti, Rudini Sampaio, Silvia Tondato

TL;DR

Results on characterizations of well-known classes of graphs via convex geometries are surveyed and some contributions to this subject are given.

Abstract

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a {\em convex geometry}. In this work we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.

Characterizations of graph classes via convex geometries: a survey

TL;DR

Results on characterizations of well-known classes of graphs via convex geometries are surveyed and some contributions to this subject are given.

Abstract

Graph convexity has been used as an important tool to better understand the structure of classes of graphs. Many studies are devoted to determine if a graph equipped with a convexity is a {\em convex geometry}. In this work we survey results on characterizations of well-known classes of graphs via convex geometries. We also give some contributions to this subject.
Paper Structure (19 sections, 32 theorems, 1 equation, 8 figures)

This paper contains 19 sections, 32 theorems, 1 equation, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An overview of graph classes and convex geometries
  4. Monophonic convexity
  5. Geodesic convexity
  6. Strong convexity
  7. $m^3$-convexity
  8. Steiner convexity
  9. Toll convexity
  10. Weakly toll convexity
  11. $l^k$-convexity
  12. Other convexities
  13. $P_3$ convexity
  14. Triangle path convexity
  15. All-path convexity
  16. ...and 4 more sections

Key Result

Theorem 1

farber-jamison A graph $G$ is chordal if and only if the monophonic convexity of $G$ is a convex geometry.

Figures (8)

  • Figure 1: Anti-exchange property applied to $S=\{a,b,c,d\}$.
  • Figure 2: A gem $G'$ with $V(G')=\{a,b,c,d,e\}$.
  • Figure 3: From left to right: hole, house, domino, $A$.
  • Figure 4: From left to right: claw, paw, $P_4$. Central vertices are indicated by arrows.
  • Figure 5: Graphs $G_1, G_2, G_3, G_4$.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • proof
  • Theorem 9
  • ...and 30 more