Nearly all cacti are edge intersection hypergraphs of 3-uniform hypergraphs

TL;DR

This paper demonstrates that almost all cacti graphs can be represented as edge intersection hypergraphs derived from 3-uniform hypergraphs, expanding understanding of hypergraph representations.

Contribution

It proves that nearly all cacti are edge intersection hypergraphs of 3-uniform hypergraphs using clique-fusion and known characterizations.

Findings

Most cacti are edge intersection hypergraphs of 3-uniform hypergraphs

Utilizes clique-fusion technique in the proof

Builds on characterizations of trees and cycles as edge intersection hypergraphs

Abstract

If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq2\}$. Using the so-called clique-fusion, we show that nearly all cacti are edge intersection hypergraphs of 3-uniform hypergraphs. In the proof we make use of known characterizations of the trees and the cycles which are edge intersection hypergraphs of 3-uniform hypergraphs (see arXiv:1901.06292).

Full text

Mathematics Subject Classification 2010: 05C65

1 Introduction and basic definitions

All hypergraphs ${\cal H}=(V({\cal H}),{\cal E}({\cal H}))$ and graphs $G=(V(G),E(G))$ considered in the following may have isolated vertices but no multiple edges or loops.

A hypergraph ${\cal H}=(V,{\cal E})$ is $k$-uniform if all hyperedges $e\in{\cal E}$ have the cardinality $k$. Trivially, any 2-uniform hypergraph ${\cal H}$ is a graph. The degree $d(v)$ (or $d_{{\cal H}}(v)$) of a vertex $v\in V$ is the number of hyperedges $e\in{\cal E}$ being incident to the vertex $v$. In standard terminology we follow Berge [1].

If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_{1}\cap e_{2}\ |\ e_{1},e_{2}\in{\cal E}\ \wedge\ e_{1}\neq e_{2}\ \wedge\ |e_{1}\cap e_{2}|\geq 2\}$.

For an application, as well as for more details on structural properties of edge intersection hypergraphs, see [8].

Obviously, for certain hypergraphs ${\cal H}$ the edge intersection hypergraph $EI({\cal H})$ can be 2-uniform; in this case $EI({\cal H})$ is a simple, undirected graph $G$. But in contrast to the notions intersection graph or edge intersection graph (cf. [2], [3], [4], [5] and [7]), $G=EI({\cal H})$ and ${\cal H}$ have one and the same vertex set $V(G)=V({\cal H})$. Therefore we consistently use our notation "edge intersection hypergraph" also when this hypergraph is 2-uniform.

First of all, in Section 2 we introduce a powerful tool for the construction of edge intersection hypergraphs, the so-called clique-fusion.

Then we restrict our investigations to cacti. A simple, connected graph $G=(V,E)$ is referred to as a cactus if and only if every edge $e\in E$ is contained in at most one cycle of $G$. The circumference $ci(G)$ of a graph $G$ is the length of a longest cycle in $G$.

In Section 3 we describe a special decomposition of cacti into trees and cycles and make use of two important results from [8]. The first one includes that all but seven exceptional trees are edge intersection hypergraphs of 3-uniform hypergraphs. The exceptional trees have at most 6 vertices. Besides the notation $P_{n}$ for the path with $n$ vertices, in the subsequent Theorem 1 we find two additional notations, namely $T7$ and $T12$ of trees with 5 and 6 vertices, respectively. These notations come from [6], they denote the graphs

$T7=(V=\{v_{1},v_{2},\ldots,v_{5}\},E=\{\{v_{1},v_{2}\},\{v_{2},v_{3}\},\{v_{3},v_{4}\},\{v_{3},v_{5}\}\})$ and

$T12=(V=\{v_{1},v_{2},\ldots,v_{6}\},E=\{\{v_{1},v_{2}\},\{v_{2},v_{3}\},\{v_{3},v_{4}\},\{v_{2},v_{5}\},\{v_{5},v_{6}\}\})$ (see Fig.1).

Theorem 1.1**.**

(Theorem 6 in [8]). All trees but $P_{2},P_{3},P_{4},P_{5},P_{6},T7$ and $T12$ are edge intersection hypergraphs of a 3-uniform hypergraph ${\cal H}$.

Note that Theorem 1 characterizes the cycle-free cacti being edge intersection hypergraphs of 3-uniform hypergraphs. The following characterization of the cycles which are edge intersection hypergraphs of 3-uniform hypergraphs is very simple to verify.

Theorem 1.2**.**

(Corollary 1 in [8]). For $n\geq 5$ the cycle $C_{n}$ is an edge intersection hypergraph of a 3-uniform hypergraph.

Using the clique-fusion, we prove that cacti having either a circumference of at least 5 or containing (in the decomposition mentioned above) a tree $T\notin\{P_{1},P_{2},P_{3},P_{4},P_{5},P_{6},T7,T12\}$ are edge intersection hypergraphs of 3-uniform hypergraphs.

At the end of the introduction, let us mention a tool from [8], which is useful for the investigation of small examples. For this end let $G=(V,E)$ and ${\cal H}=(V,{\cal E})$ be a graph and a hypergraph, respectively, having one and the same vertex set $V$. The verification of ${\cal E}(EI({\cal H}))=E(G)$ can be done by hand or by computer, e.g. using the computer algebra system MATHEMATICA [9] with the function

$EEI[eh\_]:=Complement[Select[Union[Flatten[Outer[Intersection,eh,eh,1],1]],$

$Length[\#]>1\&],eh],$

where the argument $eh$ has to be the list of the hyperedges of ${\cal H}$ in the form $\{\{a,b,c\},\ldots,\{x,y,z\}\}$. Then $EEI[eh]$ provides the list of the hyperedges of $EI({\cal H})$.

2 The clique-fusion and trees and cycles

Let $r\geq 2$ and $G_{1}=(V_{1},E_{1}),G_{2}=(V_{2},E_{2}),\ldots,G_{r}=(V_{r},E_{r})$ be graphs. Moreover, let $k\geq 1$, $V^{\prime}=\{v_{1},\ldots,v_{k}\}:=\{v\,|\,\exists i,j\in\{1,\ldots,r\}:\,i\neq j\,\wedge\,v\in V_{i}\cap V_{j}\}$ and

$E^{\prime}:=\{\{v,v^{\prime}\}\,|\,v,v^{\prime}\in V^{\prime}\,\wedge\,\exists i\in\{1,\ldots r\}:\,\{v,v^{\prime}\}\in E_{i}\}$.

Incidentally, if for each $i\in\{1,\ldots,r\}$ the graph $G_{i}$ is connected and $V_{i}\cap V^{\prime}\neq\emptyset$, then the union $G_{1}\cup\ldots\cup G_{r}=(V_{1}\cup\ldots\cup V_{r},E_{1}\cup\ldots\cup E_{r})$ is connected, too.

Consider the case that $E^{\prime}=\{\{v_{i},v_{j}\}\,|\,1\leq i<j\leq k\}$, i.e. the subgraph $\langle V^{\prime}\rangle_{G_{1}\cup\ldots\cup G_{r}}$ induced by the vertices of $V^{\prime}$ in $G_{1}\cup\ldots\cup G_{r}$ is a $k$-clique. Then we refer to the union $G_{1}\cup\ldots\cup G_{r}$ as the clique-fusion or $k$-fusion of the graphs $G_{1}\cup\ldots\cup G_{r}$ and write $G_{1}\oplus\ldots\oplus G_{r}:=G_{1}\cup\ldots\cup G_{r}$.

For an example, consider three graphs $G_{1}=(V_{1},E_{1})$, $G_{2}=(V_{2},E_{2})$ and $G_{3}=(V_{3},E_{3})$, where $\{x,y\}\in E_{1}$, $\{y,z\}\in E_{2}$ and $\{x,z\}\in E_{3}$ are edges as well as $V_{1}\cap V_{2}=\{y\}$, $V_{2}\cap V_{3}=\{z\}$ and $V_{1}\cap V_{3}=\{x\}$ hold. Then $V^{\prime}=\{x,y,z\}$ induces a 3-clique $\langle\{x,y,z\}\rangle_{G_{1}\cup G_{2}\cup G_{3}}$ in $G_{1}\cup G_{2}\cup G_{3}$ and $G_{1}\oplus G_{2}\oplus G_{3}=G_{1}\cup G_{2}\cup G_{3}$ is the $3$-fusion of the graphs $G_{1},G_{2}$ and $G_{3}$ (see Fig.2).

Note that we obtain the same $3$-fusion taking the modified graphs $G^{\prime}_{1}=(V_{1}\cup\{z\},E_{1}\cup\{\{x,z\},\{y,z\}\})$ and $G^{\prime}_{3}=(V_{3},E_{3}\setminus\{\{x,z\}\})$ instead of $G_{1}$ and $G_{3}$, i.e. we have $G_{1}\oplus G_{2}\oplus G_{3}=G^{\prime}_{1}\oplus G_{2}\oplus G^{\prime}_{3}$.

Using the above notations we have a look at a special situation.

Special Case 1. \quad\forall i,j\in\{1,\ldots,r\}:\;V^{\prime}=V_{i}\cap V_{j}\;\wedge\;\langle V^{\prime}\rangle_{G_{i}}\mbox{ is a k-clique}.

In this case, all graphs $G_{i},G_{j}$ $(i\neq j)$ have all the vertices $v_{1},\ldots,v_{k}$ (and only these vertices) in common. Additionally, in each $G_{i}$ $(i\in\{1,\dots,k\})$ (as well as in $\langle V^{\prime}\rangle_{G_{1}\cup\ldots\cup G_{r}}$) the vertices $v_{1},\ldots,v_{k}$ induce one and the same $k$-clique.

Let us mention two further special cases; the first one corresponds to $k=1$ and the second one to $r=2$, respectively.

Special Case 2. $\quad\forall i,j\in\{1,\ldots,r\}:\;V^{\prime}=\{v\}=V_{i}\cap V_{j},$ where $v$ is a uniquely determined vertex.

Special Case 3. $\quad r=2$, i.e. we consider the clique-fusion $G_{1}\oplus G_{2}$ of two graphs.

Investigating cacti, we only need Special Case 2 and Special Case 3 in combination, i.e. we have $k=1$ as well as $r=2$. Only for proof-technical reasons, in very few exceptions we use the 2-fusion (see $G_{1}\oplus K_{1,3}$ in the part (b) of the proof of Theorem 4).

Remark 2.1**.**

The clique-fusion can be easily generalized to pairwise vertex-disjoint graphs $G_{1}=(V_{1},E_{1}),G_{2}=(V_{2},E_{2}),\ldots,G_{r}=(V_{r},E_{r})$ by identifying certain vertices $v^{i}\in V_{i}$ and $v^{j}\in V_{j}\,$: $v^{i}\equiv v^{j}$, where $1\leq i<j\leq r$. Obviously, also the edges $\{v_{1}^{i},v_{2}^{i}\}\in E_{i}$ and $\{v_{1}^{j},v_{2}^{j}\}\in E_{j}$ have to be identified ($\{v_{1}^{i},v_{2}^{i}\}\equiv\{v_{1}^{j},v_{2}^{j}\}$), if the corresponding vertices have been identified ($v_{1}^{i}\equiv v_{1}^{j}$ and $v_{2}^{i}\equiv v_{2}^{j}$).

In the following, we will only make use of the clique-fusion in its original form, not in the generalized sense described in Remark 1.

Now we prove that the clique-fusion of graphs which are edge intersection hypergraphs of 3-uniform hypergraphs is an edge intersection hypergraph of a 3-uniform hypergraph, too.

Theorem 2.2**.**

Let $G=(V,E)$ be the clique-fusion $G_{1}\oplus\ldots\oplus G_{r}$ of graphs $G_{1}=(V_{1},E_{1}),\ldots,G_{r}=(V_{r},E_{r})$, where $G_{1}=EI({\cal H}_{1}),\ldots,G_{r}=EI({\cal H}_{r})$ are edge intersection hypergraphs of the 3-uniform hypergraphs ${\cal H}_{1}=(V_{1},{\cal E}_{1}),\ldots,{\cal H}_{r}=(V_{r},{\cal E}_{r})$.

Then ${\cal H}={\cal H}_{1}\cup\ldots\cup{\cal H}_{r}$ is 3-uniform and $G=EI({\cal H})$.

Proof 2.3**.**

The 3-uniformity of ${\cal H}=(V,{\cal E})$ is trivial because of $V=V_{1}\cup\ldots\cup V_{r}$ and ${\cal E}={\cal E}_{1}\cup\ldots\cup{\cal E}_{r}$. Owing to $G=G_{1}\oplus\ldots\oplus G_{r}=G_{1}\cup\ldots\cup G_{r}=EI({\cal H}_{1})\cup\ldots\cup EI({\cal H}_{r})$ in addition to $V=V_{1}\cup\ldots\cup V_{r}$ we have $E=E_{1}\cup\ldots\cup E_{r}={\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$ and it suffices to show ${\cal E}(EI({\cal H}))={\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$. (Note that, in general, neither $V_{1},\ldots,V_{r}$ nor $E_{1}={\cal E}(EI({\cal H}_{1})),\ldots,E_{r}={\cal E}(EI({\cal H}_{r}))$ are pairwise distinct.)

*Part 1: ${\cal E}(EI({\cal H}))\supseteq{\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$**.

We consider an arbitrarily chosen edge $\{x,y\}\in{\cal E}(EI({\cal H}_{i}))$, where $i\in\{1,\ldots,r\}$. Then there are hyperedges $e_{i},e_{i}^{\prime}\in{\cal E}({\cal H}_{i})\subseteq{\cal E}({\cal H})$ with $e_{i}\cap e^{\prime}_{i}=\{x,y\}$. This implies $\{x,y\}\in{\cal E}(EI({\cal H}))$.*

*Part 2: ${\cal E}(EI({\cal H}))\subseteq{\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$**.

Let $e,e^{\prime}\in{\cal E}({\cal H})$ with $e\cap e^{\prime}\in{\cal E}(EI({\cal H}))$. The 3-uniformity of ${\cal H}$ includes $|e\cap e^{\prime}|=2$.*

Assume, there exists an $i\in\{1,\ldots,r\}$ such that $e,e^{\prime}\in{\cal E}({\cal H}_{i})$. Then $e\cap e^{\prime}\in{\cal E}(EI({\cal H}_{i}))\subseteq{\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$.

*Otherwise, for all $i,j\in\{1,\ldots,r\}$ from $e\in{\cal E}({\cal H}_{i})$ and $e^{\prime}\in{\cal E}({\cal H}_{j})$ we get $i\neq j$. In this case we have $e\cap e^{\prime}=\{x,y\}$, with $\{x,y\}\subseteq V_{i}\cap V_{j}$ and $x\neq y$. Since $G$ is the clique-fusion $G_{1}\oplus\ldots\oplus G_{r}$, the vertices $x$ and $y$ have to be adjacent in $G$ and, therefore, $e\cap e^{\prime}=\{x,y\}\in E(G)=E=E_{1}\cup\ldots\cup E_{r}={\cal E}(EI({\cal H}_{1}))\cup\ldots\cup{\cal E}(EI({\cal H}_{r}))$.

(Note that because of $e\cap e^{\prime}=\{x,y\}\in E_{1}\cup\ldots\cup E_{r}$ there exist an $l\in\{1,\ldots,r\}$ and hyperedges $e_{l},e_{l}^{\prime}\in{\cal E}({\cal H}_{l})$ with $e_{l}\cap e_{l}^{\prime}=\{x,y\}$. Therefore, we even get $e\cap e^{\prime}=e_{l}\cap e_{l}^{\prime}\in{\cal E}(EI({\cal H}_{l}))=E_{l}$.) *

Let the graph $G_{1}$ be the edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{1}$. Besides the clique-fusion of $G_{1}$ and a graph $G_{2}$ being the edge intersection hypergraph of a second 3-uniform hypergraph ${\cal H}_{2}$ (see Theorem 3), we will need also the clique-fusion of $G_{1}$ with arbitrary trees (cf. Theorem 4) and cycles (cf. Theorem 5), respectively.

Theorem 2.4**.**

Let the graph $G_{1}=(V_{1},E_{1})$ be the edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{1}=(V_{1},{\cal E}_{1})$, $v\in V_{1}$ with $d_{G_{1}}(v)\geq 2$ and $T=(V_{2},E_{2})$ be a tree with $V_{2}=\{v_{1},v_{2},\ldots,v_{n}\}$, where $n\geq 2$. Moreover, let $V_{1}\cap V_{2}=\{v\}$. Then the 1-fusion $G_{1}\oplus T$ is an edge intersection hypergraph of a 3-uniform hypergraph.

Proof 2.5**.**

If $T$ is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{2}=(V_{2},{\cal E}_{2})$, owing to Theorem 3 there is nothing to show. So we have to consider only the exceptional trees $P_{2},P_{3},P_{4},P_{5},P_{6},T7$ and $T12$ from Theorem 1. Note that the one and only case where the condition $d_{G_{1}}(v)\geq 2$ will be needed is the path $P_{2}$. At first we investigate the paths $P_{n}=(V_{2},E_{2})$ with $E_{2}=\{\{v_{1},v_{2}\},\ldots,\{v_{n-1},v_{n}\}\}.$

In advance, we mention that $|V_{1}|\geq 4$ is valid, since $G_{1}$ is an edge intersection hypergraph of a 3-uniform hypergraph contaning the non-isolated vertex $v\in V_{1}$.

1. (a)

$T=P_{2}$**.

---

Let $v=v_{1}$ and $u,w\in V_{1}$ be two neighbors of $v_{1}$ in $G_{1}$. Then $G_{1}\oplus P_{2}=EI({\cal H})$ with ${\cal H}=(V_{1}\cup\{v_{2}\},{\cal E}_{1}\cup\{\{u,v_{1},v_{2}\},\{w,v_{1},v_{2}\}\})$.* 2. (b)

$T\in\{P_{3},P_{4},P_{5},P_{6}\}$**.

---

For $3\leq n\leq 6$, let $i\leq n$, $v=v_{i}$ and $u,w\in V_{1}$, where $u$ is a neighbor of $v_{i}$ in $G_{1}$.*

First we consider the situation $n=3$ and $v=v_{2}$, i.e. $v$ is the inner vertex of $P_{3}$. Then the 1-fusion $G_{1}\oplus P_{3}$ is nothing else than the 2-fusion $G_{1}\oplus K_{1,3}$, where $K_{1,3}=(V_{2}\cup\{u\},E_{2}\cup\{\{v_{2},u\}\})$. Theorem 3 implies that this $2$-fusion of $G_{1}$ and $K_{1,3}$ has the required properties.

Next we investigate the $1$-fusion $G_{1}\oplus P_{n}$ in the end vertex $v=v_{1}$ of $P_{n}$.

For this end, we consider the hypergraph ${\cal H}=(V_{1}\cup\{v_{2},\ldots,v_{n}\},{\cal E}_{1}\cup\{\{u,v_{1},v_{2}\},\{v_{1},v_{2},v_{3}\},$ $\{v_{2},v_{3},v_{4}\},\ldots,\{v_{n-2},v_{n-1},v_{n}\},\{w,v_{n-1},v_{n}\}\})$ and obtain $G_{1}\oplus P_{n}=EI({\cal H})$.

The remaining cases ($n\geq 4$* and the vertex $v_{i}\in V_{1}\cap V_{2}$ is an inner vertex of the path $P_{n}$, i.e. $1<i<n$) can be obtained from the results above in two steps.*

Let $P^{\prime}_{n}=(V^{\prime}_{2}=\{v_{1},v_{2},\ldots,v_{i}\},E^{\prime}_{2}=\{\{v_{1},v_{2}\},\{v_{2},v_{3}\},\ldots,\{v_{i-1},v_{i}\}\})$ and $P^{\prime\prime}_{n}=(V^{\prime\prime}_{2}=\{v_{i},v_{i+1},\ldots,v_{n}\},E^{\prime}_{2}=\{\{v_{i},v_{i+1}\},\{v_{i+1},v_{i+2}\},\ldots,\{v_{n-1},v_{n}\}\})$. Assume, $|V^{\prime}_{2}|\geq|V^{\prime\prime}_{2}|$. Then $P^{\prime}_{n}$ contains at least three vertices and the assumption of case (b) is fulfilled. In a first step, $G_{1}\oplus P^{\prime}_{n}$ and, in a second step, $(G_{1}\oplus P^{\prime}_{n})\oplus P^{\prime\prime}_{n}$ is an edge intersection hypergraph of a 3-uniform hypergraph, respectively. Only in the second step, when $n-i=1$ holds (i.e. $P^{\prime\prime}_{n}$ contains exactly two vertices) we have to make use of part (a). In this case, the vertex $v_{i}$ has minimum degree 2 in $G_{1}\oplus P^{\prime}_{n}$; hence part (a) is applicable.

Now we come to the 1-fusion of $G_{1}$ and the exceptional trees $T7$ and $T12$.

1. (c)

$T\in\{T7,T12\}$**.

---

Trivially, every vertex in $V_{2}=V(T)$ is included in a path of length 4 in $T$. So let $P_{4}$ be such a path in $T$ containing the vertex $v\in V_{1}\cap V_{2}$. Obviously, $T$ is a 1-fusion of $P_{4}$ and a second path $P_{t}$, such that $T=P_{4}\oplus P_{t}$ and $V(P_{4})\cap V(P_{t})=\{v^{\prime}\}$ for a certain vertex $v^{\prime}\in V_{2}$ with $d_{T}(v^{\prime})=3$. Clearly, for $T=T7$ we have $t=2$ and for $T=T12$ we get $t=3$.*

Part (b) provides that the 1-fusion $G_{1}\oplus P_{4}$ is an edge intersection hypergraph of a 3-uniform hypergraph. In order to obtain the final 1-fusion $G_{1}\oplus T$ from $G_{1}\oplus P_{4}$ it suffices to add the path $P_{t}$, i.e. $G_{1}\oplus T=(G_{1}\oplus P_{4})\oplus P_{t}$. In dependence on $t$, part (a) and part (b), respectively, provides that $G_{1}\oplus T$ is an edge intersection hypergraph of a 3-uniform hypergraph.

*Note that the assumption for (a) is fulfilled, since $d_{G_{1}\oplus P_{4}}(v^{\prime})\geq d_{T}(v^{\prime})-1=2$, i.e. the vertex $v^{\prime}$ has at least two neighbours in $G_{1}\oplus P_{4}$. ***

Remark 2.6**.**

Because in the above proof the condition $d_{G_{1}}(v)\geq 2$ is needed only for $T=P_{2}$, this condition in Theorem 4 can be weakened to $d_{G_{1}}(v)\geq 1$ if we restrict ourselves on trees with at least 3 vertices.

Theorem 2.7**.**

Let the graph $G_{1}=(V_{1},E_{1})$ be the edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{1}=(V_{1},{\cal E}_{1})$, $v\in V_{1}$ with $d_{G_{1}}(v)\geq 1$ and $C_{n}=(V_{2},E_{2})$ be the cycle of length $n\geq 3$. Moreover, let $V_{1}\cap V_{2}=\{v\}$ and, for $n=4$, the number of vertices in $G_{1}$ be at least 5. Then the 1-fusion $G_{1}\oplus C_{n}$ is an edge intersection hypergraph of a 3-uniform hypergraph.

Proof 2.8**.**

For $n\geq 5$, Theorem 2 provides that $C_{n}=(V_{2},E_{2})$ is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{2}=(V_{2},{\cal E}_{2})$. Therefore, owing to Theorem 3 there is nothing to show in this case.

So we have to investigate only the cycles $C_{3}$ and $C_{4}$. For this end let $V_{2}=\{v_{1},v_{2},\ldots,v_{n}\}$, $E_{2}=\{\{v_{1},v_{2}\},\ldots,\{v_{n-1},v_{n}\},\{v_{n},v_{1}\}\}$ and $v_{1}=v\in V_{1}\cap V_{2}$. Moreover, let $u\in V_{1}$ be a neighbor of $v_{1}$ in the graph $G_{1}$. As mentioned at the beginning of the proof of Theorem 4, we have $|V_{1}|\geq 4$ and so we can choose a vertex $x\in V_{1}\setminus\{v_{1},u\}$.

1. (a)

$n=3$**.

---

We consider ${\cal H}=(V_{1}\cup\{v_{2},v_{3}\},{\cal E}_{1}\cup\{\{v_{1},v_{2},v_{3}\},\{u,v_{1},v_{2}\},\{u,v_{1},v_{3}\},\{x,v_{2},v_{3}\}\})$.*

*In comparison with $G_{1}$, the new hyperedges $\{v_{1},v_{2},v_{3}\},\{u,v_{1},v_{2}\},\{u,v_{1},v_{3}\},$ $\{x,v_{2},v_{3}\}$ induce no additional edges in $(G_{1}\oplus C_{3})\setminus\{v_{2},v_{3}\}$: *

Clearly, $\{u,v_{1},v_{2}\}\cap\{u,v_{1},v_{3}\}=\{u,v_{1}\}$ is always an edge in $G_{1}$, namely $\{u,v_{1}\}=\{u,v\}\in E_{1}$. Moreover, the hyperedge $\{v_{1},v_{2},v_{3}\}$ and $\{x,v_{2},v_{3}\}$ has only one vertex with $G_{1}$ in common, namely the vertex $v_{1}$ and $x$, respectively.

Hence, we obtain $G_{1}\oplus C_{3}=EI({\cal H})$. 2. (b)

$n=4$**.

---

Now we have $|V_{1}|\geq 5$ and there are two additional vertices $y,z\in V_{1}\setminus\{v_{1},u,x\}$ such that $v_{1},u,x,y,z$ are pairwise distinct. *

*We use the hypergraph ${\cal H}=(V_{1}\cup\{v_{2},v_{3},v_{4}\},{\cal E}({\cal H}))$ with *

${\cal E}({\cal H}))={\cal E}_{1}\cup\{\{v_{1},v_{2},v_{3}\},\{v_{1},v_{2},v_{4}\},\{u,v_{1},v_{4}\},\{x,v_{2},v_{3}\},\{y,v_{3},v_{4}\},\{z,v_{3},v_{4}\}\}$.

*Analogously to the previous case, it can be verified that $G_{1}\oplus C_{4}=EI({\cal H})$ holds. ***

3 Cacti as edge intersection hypergraphs

The basic idea is to decompose a given cactus $G$ into cycles and (in a certain sense maximal) trees. After that we begin with a cycle $C_{n}$ of length $n\geq 5$ and a tree $T\notin\{P_{1},P_{2},\ldots,P_{6},T7,T12\}$, respectively, (which is an edge intersection hypergraph of a 3-uniform hypergraph) and reconstruct the original cactus step by step using Theorem 4 and Theorem 5. For this purpose, in each step we build a 1-fusion of a (connected) subgraph $G_{i}$ (which is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{i}$) of the original cactus and one of the cycles or trees described above.

We begin with the decomposition. First of all, let $G=(V,E)$ be a cactus and $V^{\prime}=\{v_{1},\ldots,v_{s}\}\subseteq V$ be the set of all vertices having a degree of at least 3 and being contained in a cycle of $G$. Evidently, the vertices in $V^{\prime}$ are articulation vertices of $G$. We refer to these vertices as the decomposition vertices or shortly d-vertices of $G$. It is easy to see that $V^{\prime}=\emptyset$ is equivalent to the case that the cactus $G$ is a cycle or a tree, respectively. In this case Theorem 1 and Theorem 2 include the corresponding characterizations. So in the following assume $V^{\prime}\neq\emptyset$.

The so-called tree-cycle-decomposition of the cactus $G$ will be carried out in two consecutive steps.

Step 1. In each cycle of $G$, we delete all edges and all vertices of degree 2. Thus we obtain a forest consisting of pairwise vertex-disjoint trees $T^{1},T^{2},\ldots,T^{k}$, the so-called limbs of $G$. Clearly, every limb contains at least one $d$-vertex. If the tree $T^{j}$ is a single vertex, i.e. $V(T^{j})=\{v_{j}\}$, then $v_{j}$ is an articulation vertex that connects $\frac{1}{2}d_{G}(v_{j})$ cycles in $G$. Note that the limbs are the "(in a certain sense maximal) trees" mentioned at the beginning of the section.

Step 2. We start again with the original cactus $G$ and delete all edges $e\in E(T^{1}),\ldots,E(T^{k})$. Besides several isolated vertices, this leads to a system of Eulerian graphs which can be uniquely decomposed into a system $C^{1},\ldots,C^{l}$ of pairwise edge-disjoint cycles. We denote $C^{1},\ldots,C^{l}$ as the cycles of $G$.

The next remark is a collection of some simple, but useful properties of the tree-cycle-decomposition.

Remark 3.1**.**

1. (i)

The $d$-vertices $v_{1},\ldots,v_{s}$, the limbs $T^{1},\ldots,T^{k}$ as well as the cycles $C^{1},\ldots,C^{l}$ are uniquely determined. 2. (ii)

The limbs $T^{1},\ldots,T^{k}$ are pairwise vertex-disjoint. 3. (iii)

Any $d$-vertex is contained in at least one of the cycles of $G$. 4. (iv)

Any $d$-vertex is contained in at most one of the limbs of $G$. 5. (v)

If $V^{\prime}\neq\emptyset$, then every cycle and every limb includes at least one $d$-vertex.

Now we are ready to formulate our main theorem.

Theorem 3.2**.**

Let $G=(V,E)$ be a cactus with

1. (a)

circumference $ci(G)\geq 5$  or 2. (b)

$G$* contains a limb $T\notin\{P_{1},P_{2},\ldots,P_{6},T7,T12\}$.*

Then $G$ is an edge intersection hypergraph of a 3-uniform hypergraph.

Proof 3.3**.**

Let $\{G_{1},G_{2},\ldots,G_{k+l}\}=\{C^{1},C^{2},\ldots,C^{l},T^{1},T^{2},\ldots,T^{k}\}$, where $C^{1},C^{2},\ldots,C^{l}$ and $T^{1},T^{2},\ldots,T^{k}$ are the cycles and the limbs of $G$, respectively.

Case (a): $ci(G)\geq 5$.**

Let the indices of the subgraphs $G_{1},\ldots,G_{k+l}$ be chosen so that $G_{1}=C^{1}$ is a cycle of a length of at least 5 and, for every $p\in\{1,2,\ldots,k+l-1\}$, the subgraph $G_{1}\cup\ldots\cup G_{p}$ of the cactus $G$ has a vertex $v_{p}$ in common with the subgraph $G_{p+1}$. Obviously, \big{(}V(G_{1})\cup\ldots\cup V(G_{p})\big{)}\cap V(G_{p+1})=\{v_{p}\}\subseteq V^{\prime}, where $V^{\prime}$ is the set of the $d$-vertices of $G$.

Trivially, $G=G_{1}\cup\ldots\cup G_{k+l}$ and, for any $p\in\{1,2,\ldots,k+l-1\}$, from $|V(G_{p+1})|=1$ (this is the case if and only if $G_{p+1}$ is a trivial tree containing only one vertex, i.e. $V(G_{p+1})=\{v_{p}\}$) it follows $(G_{1}\cup\ldots\cup G_{p})\cup G_{p+1}=G_{1}\cup\ldots\cup G_{p}$. Owing to our indexing of $G_{1},G_{2},\ldots,G_{k+l}$ this is equivalent to $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})\oplus G_{p+1}=(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})$. Note that all the clique-fusions in this expression are 1-fusions and, additionally, all these clique-fusions are connected.

To mention the most trivial case, the 1-fusion of any graph $G^{\prime}=(V(G^{\prime}),E(G^{\prime}))$ with a graph $G^{\prime\prime}=(V(G^{\prime\prime})=\{v^{\prime}\},E(G^{\prime\prime})=\emptyset)$, with $v^{\prime}\in V(G^{\prime})$, is the original graph $G^{\prime}$ itself. Therefore, if $G^{\prime}$ is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}$, then also $G^{\prime}\oplus G^{\prime\prime}=EI({\cal H})$ is valid.

According to the assumption, $G_{1}$ is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{1}$ containing at least 5 vertices. Therefore $G_{1}$ fulfills also the assumptions of Theorems 4 and 5 and $G_{1}\oplus G_{2}$ is also an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{2}$. Since $G_{1}$ is a cycle, all of its vertices have degree $2$ and it plays no role whether or not $G_{2}$ is a cycle or a limb.

With two little additional arguments we can argue in the same manner for arbitrarily chosen $p\in\{2,\ldots,k+l-1\}$ considering $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})\oplus G_{p+1}$.

First, if $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})$ is an edge intersection hypergraph of a 3-uniform hypergraph ${\cal H}_{p}$ then it contains at least 5 vertices (since $|V(G_{1})|\geq 5$ holds). This implies that the clique-fusion $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})\oplus G_{p+1}$ also in the case $G_{p+1}=C_{4}$ is an edge intersection hypergraph of a 3-uniform hypergraph. Clearly, also for $G_{p+1}=C_{3}$ no problem occurs.

*Secondly, in case of $G_{p+1}=P_{2}$, we have to ensure that the vertex

v_{p}\in\big{(}V(G_{1})\cup\ldots\cup V(G_{p})\big{)}\cap V(G_{p+1}) has a degree $d_{G_{1}\cup\ldots\cup G_{p}}(v_{p})\geq 2$ (cf. Theorem 4 and Remark 2). This yields from the definition of the $d$-vertices and the limbs as well as from the construction of our tree-cycle-decomposition of the cactus $G$ in the following way. In case of $G_{p+1}=P_{2}$ the graph $G_{p+1}$ is a limb. Remark 3(iv) includes that the $d$-vertex $v_{p}\in V(G_{p+1}$ cannot be contained in another limb of $G$. Therefore $v_{p}$ is included in a cycle of $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})=G_{1}\cup\ldots\cup G_{p}$ and has a degree $d_{G_{1}\cup\ldots\cup G_{p}}(v_{p})\geq 2$.*

This completes the proof of Case (a).

Case (b): $ci(G)\leq 4$.**

We use nearly the same argumentation as in Case (a), the only modification is that we have to start with $G_{1}=T^{1}$ instead of $G_{1}=C^{1}$, where $T^{1}\notin\{P_{1},P_{2},\ldots,P_{6},T7,T12\}$ is a limb of $G$. If $G=T^{1}$ holds then there is nothing to show.

So assume $k+l>1$ and choose the indices of $G_{2},G_{3},\ldots,G_{k+l}$ in the same way as in Case (a). Thus the graph $G_{1}=T^{1}$ is again an edge intersection hypergraph of a 3-uniform hypergraph. Moreover, $G_{2}=C_{3}$ or $G_{2}=C_{4}$ and it suffices to prove that any fusion $G_{1}\oplus G_{2}=T^{1}\oplus C_{n}$ ($n\in\{3,4\}$) is an edge intersection hypergraph of a 3-uniform hypergraph. The rest of the argumentation can be taking over word-for-word from Case (a). So let us consider $T^{1}\oplus C_{n}$.

Since $T^{1}$ is an edge intersection hypergraph of a 3-uniform hypergraph, we can apply Theorem 5. The only exception is the case $|V(T^{1})|=4$ and $n=4$. Obviously, this corresponds to $T^{1}=K_{1,3}$ and we have to investigate the two possible 1-fusions of $K_{1,3}$ and $C_{4}$.

For this end, let $K_{1,3}=(V_{1}=\{v_{1},v_{2},v_{3},v_{4}\},E_{1})$, $C_{4}=(V_{2}=\{v_{4},v_{5},v_{6},v_{7}\},E_{2})$, where $E_{2}=\{\{v_{4},v_{5}\},\{v_{5},v_{6}\},\{v_{6},v_{7}\},\{v_{7},v_{4}\}\})$, and look at $K_{1,3}\oplus C_{4}$. In the following, we discuss both 1-fusions.

*The first 1-fusion we have to investigate is the situation that $v_{4}$ is not the center of the star; so w.l.o.g. let $v_{1}$ be the center, i.e. $E_{1}=\{\{v_{1},v_{2}\},\{v_{1},v_{3}\},\{v_{1},v_{4}\}\}$. Then the hypergraph

${\cal H}=(V,{\cal E})$ with ${\cal E}=\{\{v_{1},v_{2},v_{3}\},\{v_{1},v_{2},v_{4}\},\{v_{1},v_{3},v_{4}\},\{v_{1},v_{4},v_{7}\},\{v_{1},v_{5},v_{6}\},\{v_{2},v_{6},v_{7}\},$

$\{v_{3},v_{6},v_{7}\},\{v_{4},v_{5},v_{6}\},\{v_{4},v_{5},v_{7}\}\}$ provides the 1-fusion $EI({\cal H})=K_{1,3}\oplus C_{4}=(V,E)$ with $V=V_{1}\cup V_{2}$ and $E=\{\{v_{1},v_{2}\},\{v_{1},v_{3}\},\{v_{1},v_{4}\},\{v_{4},v_{5}\},\{v_{5},v_{6}\},\{v_{6},v_{7}\},\{v_{7},v_{4}\}\}$ (see the first picture in Fig. 3).*

*Now let $v_{4}$ be the center of the star. Thus we have $E_{1}=\{\{v_{1},v_{4}\},\{v_{2},v_{4}\},\{v_{3},v_{4}\}\}$. We consider the hypergraph ${\cal H}=(V,{\cal E})$ with ${\cal E}=\{\{v_{1},v_{3},v_{4}\},\{v_{1},v_{4},v_{7}\},\{v_{2},v_{3},v_{4}\},\{v_{2},v_{4},v_{5}\},$

$\{v_{2},v_{4},v_{7}\},\{v_{3},v_{6},v_{7}\},\{v_{4},v_{5},v_{6}\},\{v_{5},v_{6},v_{7}\}\}$. ${\cal H}$ has the edge intersection hypergraph $EI({\cal H})=K_{1,3}\oplus C_{4}=(V,E)$ with $V=V_{1}\cup V_{2}$ and $E=\{\{v_{1},v_{4}\},\{v_{2},v_{4}\},\{v_{3},v_{4}\},\{v_{4},v_{5}\},\{v_{5},v_{6}\},\{v_{6},v_{7}\},$ $\{v_{7},v_{4}\}\}.$ This is the second possible 1-fusion of $K_{1,3}$ and $C_{4}$ (see the second picture in Fig. 3).*

4 Concluding remarks

In Section 3 we made use of very special clique-fusions, namely the (iterated) 1-fusion of edge intersection hypergraphs of 3-uniform hypergraphs. Since cacti can be decomposed into limbs and cycles using the $d$-vertices, the 1-fusion had been proved to be a suitable tool for our investigations on cacti.

Besides cycles and trees also other classes of graphs are known to be edge intersection hypergraphs of 3-uniform hypergraphs, e.g. as wheels and complete graphs with at least 5 and 4 vertices, respectively. So it would be interesting to ask for more classes of graphs being edge intersection hypergraphs which can be constructed from known such graphs using the clique-fusion. Besides the 1-fusion also the 2-fusion seems to be a relatively simple tool for such constructions.

Remember that in Section 3 we restricted ourselves on the iterated clique-fusion in the sense, that in each step we applied the 1-fusion only to two graphs (see the proof of Theorem 6 where we dealt with the clique fusion in the form $(\ldots((G_{1}\oplus G_{2})\oplus G_{3})\ldots\oplus G_{p})\oplus G_{p+1}$. Another interesting topic for further investigations could be the usage of the $k$-fusion in a more common sense, namely as $G_{1}\oplus\ldots\oplus G_{r}$, where $r>2$ holds (see the original definition at the beginning of Section 2). We conjecture that – using the clique-fusion in this common sense – the construction of corresponding classes of edge intersection hypergraphs of 3-uniform hypergraphs may be much more complicated.

Finally, we formulate three problems.

Problem 4.1**.**

Using the clique-fusion, find more classes of edge intersection hypergraphs of 3-uniform hypergraphs.

Problem 4.2**.**

Besides the clique-fusion, find other tools for the construction of graphs being edge intersection hypergraphs of $r$-uniform $(r\geq 3)$ or non-uniform hypergraphs.

Problem 4.3**.**

Find more classes of graphs being edge intersection hypergraphs of $r$-uniform $(r\geq 3)$ or non-uniform hypergraphs.

Instead of graphs, also the investigation of hypergraphs which are edge intersection hypergraphs of certain hypergraphs is an attractive topic.