Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether the same thing holds for "graph-like" hypergraphs.

We call a hypergraph $H=(V,E)$ with $V\neq \emptyset$ connected if for all non-empty $X\subseteq V$ with $X\neq V$ there is $e\in E$ with $$e\cap X \neq \emptyset \neq e \cap (V\setminus X).$$

(Trivially, for every connected hypergraph, we have $\bigcup E = V$.)

We say that $H=(V,E)$ is linear if the cardinality of the intersection of any two distinct edges is at most $1$.

Question. If $H =(V,E)$ is a linear connected hypergraph, is there $E_0\subseteq E$ with the following properties?

1. $(V, E_0)$ is connected, and
2. whenever $e_0\in E_0$, the hypergraph $\big(V, (E_0\setminus\{e_0\})\big)$ is no longer connected.

Note. There is an easy example showing that if we consider all connected hypergraphs, the answer is negative.

Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether the same thing holds for "graph-like" hypergraphs.

Formulation for hypergraphs. We call a hypergraph $H=(V,E)$ with $V\neq \emptyset$ connected if for all non-empty $X\subseteq V$ with $X\neq V$ there is $e\in E$ with $$e\cap X \neq \emptyset \neq e \cap (V\setminus X).$$

(Trivially, for every connected hypergraph, we have $\bigcup E = V$.)

We say that $H=(V,E)$ is linear if the cardinality of the intersection of any two distinct edges is at most $1$.

Question. If $H =(V,E)$ is a linear connected hypergraph, is there $E_0\subseteq E$ with the following properties?

1. $(V, E_0)$ is connected, and
2. whenever $e_0\in E_0$, the hypergraph $\big(V, (E_0\setminus\{e_0\})\big)$ is no longer connected.

Note. There is an easy example showing that if we consider all connected hypergraphs, the answer is negative.

Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether the same thing hold for "graph-like" hypergraphs.

We call a hypergraph $H=(V,E)$ with $V\neq \emptyset$ connected if for all non-empty $X\subseteq V$ with $X\neq V$ there is $e\in E$ with $$e\cap X \neq \emptyset \neq e \cap (V\setminus X).$$

(Trivially, for every connected hypergraph, we have $\bigcup E = V$.)

We say that $H=(V,E)$ is linear if the cardinality of the intersection of any two distinct edges is at most $1$.

Question. If $H =(V,E)$ is a linear connected hypergraph, is there $E_0\subseteq E$ with the following properties?

1. $(V, E_0)$ is connected, and
2. whenever $e_0\in E_0$, the hypergraph $\big(V, (E_0\setminus\{e_0\})\big)$ is no longer connected.

Note. There is an easy example showing that if we consider all connected hypergraphs, the answer is negative.

Starting point. For every simple, undirected graph $G=(V,E)$ there is $E_0\subseteq E$ such that $(V,E_0)$ is minimally connected: the spanning tree. The goal of this question is to find out whether the same thing holds for "graph-like" hypergraphs.

We call a hypergraph $H=(V,E)$ with $V\neq \emptyset$ connected if for all non-empty $X\subseteq V$ with $X\neq V$ there is $e\in E$ with $$e\cap X \neq \emptyset \neq e \cap (V\setminus X).$$

(Trivially, for every connected hypergraph, we have $\bigcup E = V$.)

We say that $H=(V,E)$ is linear if the cardinality of the intersection of any two distinct edges is at most $1$.

Question. If $H =(V,E)$ is a linear connected hypergraph, is there $E_0\subseteq E$ with the following properties?

1. $(V, E_0)$ is connected, and
2. whenever $e_0\in E_0$, the hypergraph $\big(V, (E_0\setminus\{e_0\})\big)$ is no longer connected.

Note. There is an easy example showing that if we consider all connected hypergraphs, the answer is negative.