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More on discrete convexity

Vladimir Gurvich, Mariya Naumova

Abstract

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form.

More on discrete convexity

Abstract

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form.
Paper Structure (58 sections, 14 theorems, 2 equations, 10 figures)

This paper contains 58 sections, 14 theorems, 2 equations, 10 figures.

Table of Contents

  1. Hereditary and Convex Discrete Families
  2. Graphs and digraphs
  3. Definitions and preliminaries
  4. Connected graphs
  5. Order $\succ_V$
  6. Order $\succ_E$
  7. Disconnected graphs
  8. Order $\succ_V$
  9. Order $\succ_E$
  10. Strongly connected directed graphs
  11. Order $\succ_V$
  12. Order $\succ_E$
  13. Not strongly connected directed graphs
  14. Order $\succ_V$
  15. Order $\succ_E$
  16. ...and 43 more sections

Key Result

Lemma 1

Both families are weakly hereditary. Furthermore, $(\mathcal{F}(G), \succ_V)$ (respectively, $(\mathcal{F}(G), \succ_E)$) is hereditary if and only if $n=1$ and $G_1$ is the null-graph (respectively, the edge-free graph).

Figures (10)

  • Figure 1: Wrochna's example
  • Figure 2: Icosidodecahedron. Illustration for Luca Pacioli's "Divina proportione" by Leonardo da Vinci
  • Figure 3: $2$-graph $\Pi$ and $3$-graph $\Delta$
  • Figure 4: $2$-graph $\mathcal{G} = \Pi(v_4 \to \Pi'$)
  • Figure 5: 2-graphs $\mathcal{B}$ and $\mathcal{G}$
  • ...and 5 more figures

Theorems & Definitions (35)

  • Remark 1
  • Lemma 1
  • Example 1
  • Remark 2
  • Proposition 1
  • Proposition 2
  • Example 2
  • Example 3
  • Proposition 3
  • Example 4
  • ...and 25 more