Hypo-efficient domination and hypo-unique domination

TL;DR

This paper introduces and characterizes hypo-efficient and hypo-unique domination graphs, exploring their properties, bounds, and specific families like circulant graphs, with implications for domination and bondage numbers.

Contribution

It defines new classes of hypo-domination graphs, establishes their fundamental properties, bounds their order, and identifies families achieving these bounds.

Findings

Hypo-UD graphs of order at least 3 are connected.

For hypo-UD graphs, removing any vertex reduces the domination number.

Bondage number of hypo-UD graphs is at most minimum degree plus one.

Abstract

For a graph $G$ let $\gamma (G)$ be its domination number. We define a graph G to be (i) a hypo-efficient domination graph (or a hypo-$\mathcal{ED}$ graph) if $G$ has no efficient dominating set (EDS) but every graph formed by removing a single vertex from $G$ has at least one EDS, and (ii) a hypo-unique domination graph (a hypo-$\mathcal{UD}$ graph) if $G$ has at least two minimum dominating sets,but $G-v$ has a unique minimum dominating set for each $v\in V(G)$. We show that each hypo-$\mathcal{UD}$ graph $G$ of order at least $3$ is connected and $\gamma(G-v) < \gamma(G)$ for all $v \in V(G)$. We obtain a tight upper bound on the order of a hypo-$\mathcal{P}$ graph in terms of the domination number and maximum degree of the graph, where $\mathcal{P} \in \{\mathcal{UD}, \mathcal{ED}\}$. Families of circulant graphs which achieve these bounds are presented. We also prove that the bondage number of any hypo-$\mathcal{UD}$ graph is not more than the minimum degree plus one.