Carath$é$odory Number and Exchange Number in $Δ$-convexity

TL;DR

This work develops the theory of $\Delta$-convexity on graphs by studying Carathéodory and exchange numbers and their behavior on standard graph families. It establishes sharp bounds and exact values for these invariants on various graph classes (e.g., triangle-free, complete, block graphs) and analyzes their interaction under the Cartesian, strong, and lexicographic graph products, revealing multiplicative behavior for Carathéodory numbers and tight lower bounds for exchange numbers. The results include precise values for block graphs, 2-connected chordal graphs, and constructions attaining the bounds, as well as sharpness results for products such as $G\Box H$, $G\boxtimes H$, and $G\circ H$, with notable conclusions like $c_{\Delta}(G\boxtimes H)=2$ and $e_{\Delta}(G\circ H)$ taking values 2 or 3 depending on structural properties. These findings advance understanding of convexity-like invariants in graph theory and provide groundwork for further algorithmic and structural explorations in $\Delta$-convexity and related graph operations.

Abstract

Given a graph $G$, a set is $Δ$-convex if there is no vertex $u\in V(G)\setminus S$ forming a triangle with two vertices of $S$. The $Δ$-convex hull of $S$ is the minimum $Δ$-convex set containing $S$. This article is an attempt to discuss the Carathéodory number and exchange number on various graph families and standard graph products namely Cartesian, strong and, lexicographic products of graphs.

Figures (2)

• Figure 1: Graph $G$ with with $c_\Delta(G)=e_\Delta(G)=n$.
• Figure 2: Graph $G$ with $k$ triangles and $e_\Delta(G)=k+2$.

Theorems & Definitions (39)

• Theorem 1: Sierksma Inequalities van_de_vel
• Theorem 2
• proof
• Proposition 3
• proof
• Remark 4
• Lemma 5
• proof
• Lemma 6
• Theorem 7