Incidence-free sets and edge domination in incidence graphs

TL;DR

This paper investigates the edge domination number in incidence graphs of combinatorial designs, linking it to incidence-free sets, and employs diverse mathematical techniques to analyze these properties.

Contribution

It establishes a connection between edge domination in incidence graphs and incidence-free sets in symmetric designs, expanding understanding of these combinatorial structures.

Findings

Determines the edge domination number for certain incidence graphs.

Links edge domination to the size of incidence-free sets in symmetric designs.

Uses combinatorial, probabilistic, geometric, and spectral methods for analysis.

Abstract

A set of edges $\Gamma$ of a graph $G$ is an edge dominating set if every edge of $G$ intersects at least one edge of $\Gamma$, and the edge domination number $\gamma_e(G)$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study $\gamma_e(G)$ for graphs $G$ which are the incidence graph of some incidence structure $D$, with an emphasis on the case when $D$ is a symmetric design. In particular, we show in this latter case that determining $\gamma_e(G)$ is equivalent to determining the largest size of certain incidence-free sets of $D$. Throughout, we employ a variety of combinatorial, probabilistic and geometric techniques, supplemented with tools from spectral graph theory.