Modulus of edge covers and stars

TL;DR

This paper initiates the exploration of the modulus of edge covers, establishing theoretical results and demonstrating, by numerical experiments, some interesting behaviors of this modulus.

Abstract

This paper explores the modulus (discrete $p$-modulus) of the family of edge covers on a discrete graph. This modulus is closely related to that of the larger family of fractional edge covers; the modulus of the latter family is guaranteed to approximate the modulus of the former within a multiplicative factor. The bounds on edge cover modulus can be computed efficiently using a duality result that relates the fractional edge covers to the family of stars.

Figures (11)

• Figure 1: A standard set of graphs for modulus examples.
• Figure 2: A minimal edge cover for an odd cycle $C$.
• Figure 5: A path that returns to an odd cycle $C$ and creates an even cycle, i.e. the "snowman" graph. Orange edges make up the even cycles.
• Figure 6: Values of $\tau$ for two connected cycles.
• Figure 7: Extreme points of $\Gamma_{\text{fec}}$ (up to rotation and reflection) for $G = W_5$. There are 25 extreme points in total.

Theorems & Definitions (54)

• Lemma 2.1
• proof
• Proposition 2.2: Monotonicity
• Lemma 2.3
• proof
• Example 1: Star Graph
• Lemma 3.1
• proof
• Example 2: Cycle Graph
• Example 3: Complete Graph