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Separation axiom $S_3$ for geodesic convexity in graphs

Victor Chepoi

TL;DR

This work analyzes the separation axiom $S_3$ for geodesic convexity in graphs, characterizing $S_3$-graphs via shadows and maximal $x_0$-proximal sets and relating semispaces to shadows $K/x_0$ and $x_0/K$. It gives a detailed structural theory, showing that in TC-satisfying graphs, maximal proximal sets correspond to pointed maximal cliques, enabling polynomial-time enumeration of semispaces; in the broader meshed class, $S_3$-graphs are precisely meshed graphs excluding five forbidden subgraphs. The paper further proves that in meshed graphs, local convexity implies global convexity and establishes fiber-complemented decompositions, with applications to matroid basis graphs and various planar and partial cube classes. It also tackles the NP-hard halfspace separation problem, offering two solution pathways and applying them to specific graph-convexities, thus bridging convexity theory with algorithmic and complexity considerations. Overall, the results provide a comprehensive, structure-driven framework for $S_3$ in graphs, yielding both theoretical insight and practical algorithms for semispace enumeration and separation.

Abstract

Semispaces of a convexity space $(X,C)$ are maximal convex sets missing a point. The separation axiom $S_3$ asserts that any point $x_0\in X$ and any convex set $A$ not containing $x_0$ can be separated by complementary halfspaces (convex sets with convex complements) or, equivalently, that all semispaces are halfspaces. In this paper, we study $S_3$ for geodesic convexity in graphs and the structure of semispaces in $S_3$-graphs. We characterize $S_3$-graphs and their semispaces in terms of separation by halfspaces of vertices $x_0$ and special sets, called maximal $x_0$-proximal sets and in terms of convexity of their mutual shadows $x_0/K$ and $K/x_0$. In $S_3$-graphs $G$ satisfying the triangle condition (TC), maximal proximal sets are the pre-maximal cliques of $G$ (i.e., cliques $K$ such that $K\cup\{ x_0\}$ are maximal cliques). This allows to characterize the $S_3$-graphs satisfying (TC) in a structural way and to enumerate their semispaces efficiently. In case of meshed graphs (an important subclass of graphs satisfying (TC)), the $S_3$-graphs have been characterized by excluding five forbidden subgraphs. On the way of proving this result, we also establish some properties of meshed graphs, which maybe of independent interest. In particular, we show that any connected, locally-convex set of a meshed graph is convex. We also provide several examples of $S_3$-graphs, including the basis graphs of matroids. Finally, we consider the (NP-complete) halfspace separation problem, describe two methods of its solution, and apply them to particular classes of graphs and graph-convexities.

Separation axiom $S_3$ for geodesic convexity in graphs

TL;DR

This work analyzes the separation axiom for geodesic convexity in graphs, characterizing -graphs via shadows and maximal -proximal sets and relating semispaces to shadows and . It gives a detailed structural theory, showing that in TC-satisfying graphs, maximal proximal sets correspond to pointed maximal cliques, enabling polynomial-time enumeration of semispaces; in the broader meshed class, -graphs are precisely meshed graphs excluding five forbidden subgraphs. The paper further proves that in meshed graphs, local convexity implies global convexity and establishes fiber-complemented decompositions, with applications to matroid basis graphs and various planar and partial cube classes. It also tackles the NP-hard halfspace separation problem, offering two solution pathways and applying them to specific graph-convexities, thus bridging convexity theory with algorithmic and complexity considerations. Overall, the results provide a comprehensive, structure-driven framework for in graphs, yielding both theoretical insight and practical algorithms for semispace enumeration and separation.

Abstract

Semispaces of a convexity space are maximal convex sets missing a point. The separation axiom asserts that any point and any convex set not containing can be separated by complementary halfspaces (convex sets with convex complements) or, equivalently, that all semispaces are halfspaces. In this paper, we study for geodesic convexity in graphs and the structure of semispaces in -graphs. We characterize -graphs and their semispaces in terms of separation by halfspaces of vertices and special sets, called maximal -proximal sets and in terms of convexity of their mutual shadows and . In -graphs satisfying the triangle condition (TC), maximal proximal sets are the pre-maximal cliques of (i.e., cliques such that are maximal cliques). This allows to characterize the -graphs satisfying (TC) in a structural way and to enumerate their semispaces efficiently. In case of meshed graphs (an important subclass of graphs satisfying (TC)), the -graphs have been characterized by excluding five forbidden subgraphs. On the way of proving this result, we also establish some properties of meshed graphs, which maybe of independent interest. In particular, we show that any connected, locally-convex set of a meshed graph is convex. We also provide several examples of -graphs, including the basis graphs of matroids. Finally, we consider the (NP-complete) halfspace separation problem, describe two methods of its solution, and apply them to particular classes of graphs and graph-convexities.
Paper Structure (43 sections, 72 theorems, 3 equations, 2 figures)

This paper contains 43 sections, 72 theorems, 3 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convexity spaces and separation axioms
  4. Interval spaces
  5. Graphs
  6. Examples of $S_3$-graphs
  7. $S_3$-Graphs
  8. $S_3$-convexity spaces
  9. Properties of $S_3$-graphs
  10. Semispaces in $S_3$-graphs
  11. Characterization of $S_3$-graphs
  12. Maximal $x_0$-proximal sets
  13. $S_3$-graphs satisfying (TC)
  14. The structure of semispaces
  15. Characterization of $S_3$-graphs
  16. ...and 28 more sections

Key Result

Theorem 1

Ca A convexity space $(X,\mathop{\mathrm{\frak C}}\nolimits)$ satisfies JHC if and only if it satisfies the Peano axiom: for any $u,v,w\in X$, $x\in \mathop{\mathrm{\frak c}}\nolimits(w,v)$, and $y\in \mathop{\mathrm{\frak c}}\nolimits(u,x)$ there exists $z\in \mathop{\mathrm{\frak c}}\nolimits(u,v)

Figures (2)

  • Figure 1: A meshed $S_3$-graph $\Gamma$ with non-convex union shadow $K|x_{0}$ with $K=\{ y,z\}$
  • Figure 2: Meshed non-$S_3$ graphs.

Theorems & Definitions (181)

  • Theorem 1
  • Definition 1: Halfspaces and separation axioms
  • Definition 2: Semispaces
  • Example 1
  • Theorem 2
  • Definition 3: Shadows
  • Remark 1
  • Theorem 3
  • Remark 2
  • Proposition 1
  • ...and 171 more