Domination in digraphs and their products

TL;DR

This paper investigates domination properties in digraphs and their products, establishing new bounds and equalities for domination numbers in specific classes like ditrees and general digraphs, with implications for graph product theory.

Contribution

The paper proves that in certain digraphs with girth at least 7, neighborhoods have the Helly property, and establishes new product formulas and bounds for domination numbers in digraphs and their products.

Findings

Helly property holds for neighborhoods in high-girth digraphs

Equality $ ho(T)= ho^{ m o}(T)= ext{domination numbers}$ in ditrees

New bounds for domination numbers in Cartesian products of digraphs

Abstract

A dominating (respectively, total dominating) set $S$ of a digraph $D$ is a set of vertices in $D$ such that the union of the closed (respectively, open) out-neighborhoods of vertices in $S$ equals the vertex set of $D$. The minimum size of a dominating (respectively, total dominating) set of $D$ is the domination (respectively, total domination) number of $D$, denoted $\gamma(D)$ (respectively,$\gamma_t(D)$). The maximum number of pairwise disjoint closed (respectively,open) in-neighborhoods of $D$ is denoted by $\rho(D)$ (respectively,$\rho^{\rm o}(D)$). We prove that in digraphs whose underlying graphs have girth at least $7$, the closed (respectively,open) in-neighborhoods enjoy the Helly property, and use these two results to prove that in any ditree $T$ (that is, a digraph whose underlying graph is a tree), $\gamma_t(T)=\rho^{\rm o}(T)$ and $\gamma(T)=\rho(T)$. By using the former equality we then prove that $\gamma_t(G\times T)=\gamma_t(G)\gamma_t(T)$, where $G$ is any digraph and $T$ is any ditree, each without a source vertex, and $G\times T$ is their direct product. From the equality $\gamma(T)=\rho(T)$ we derive the bound $\gamma(G\mathbin{\Box} T)\ge\gamma(G)\gamma(T)$, where $G$ is an arbitrary digraph, $T$ an arbitrary ditree and $G\mathbin{\Box} T$ is their Cartesian product. In general digraphs this Vizing-type bound fails, yet we prove that for any digraphs $G$ and $H$, where $\gamma(G)\ge\gamma(H)$, we have $\gamma(G \mathbin{\Box} H) \ge \frac{1}{2}\gamma(G)(\gamma(H) + 1)$. This inequality is sharp as demonstrated by an infinite family of examples. Ditrees $T$ and digraphs $H$ enjoying $\gamma(T\mathbin{\Box} H)=\gamma(T)\gamma(H)$ are also investigated.