<p>According to <a href="https://en.wikipedia.org/wiki/Butterfly_graph" rel="nofollow noreferrer">Wikipedia</a>, this graph is known variously as the <b>butterfly graph</b>, <b>bowtie graph</b>, or <b>hourglass graph</b>.</p>

CC BY-SA 4.0
2021-10-22T17:53:49.710Z
With the caveat that the butterfly/bowtie/hourglass graph is just one particular graph.  It is not a general name for any graph that can be formed as the union of triangles.
Neal Young
CC BY-SA 4.0
2021-10-22T19:30:51.643Z
Ah, indeed. I was only looking at the graph $H$ from the question. I think the OP would need to provide some more examples where there are >2 triangles in the union in order to see more clearly what family of graphs are under consideration (E.g.: are all of the triangles joined only at one common vertex or are there other ways of joining them?)
mhum
CC BY-SA 4.0
2021-10-22T16:30:58.527Z
2021-10-22T19:28:28.423Z
2021-10-22T19:28:28.423Z
mhum
<p>I have different simple triangle graphs. As an instance, <span class="math-container">$G_1=(V_1,E_1)=(\{1,2,3\},\{\{1,2\},\{2,3\},\{3,1\}\})$</span> and <span class="math-container">$G_2=(V_2,E_2)=(\{1,4,5\},\{\{1,4\},\{4,5\},\{5,1\}\})$</span>.</p>
<p>The union of both graphs is <span class="math-container">$H=(V_1 \cup V_2, E_1 \cup E_2)=(\{1,2,3,4,5\},\{\{1,2\},\{2,3\},\{3,1\},\{1,4\},\{4,5\},\{5,1\}\})$</span></p>
<p>Does a graph like <span class="math-container">$H$</span> that results from the union of triangles have a particular name or properties?</p>

CC BY-SA 4.0
2021-10-22T17:54:38.233Z
(i) You are looking for a name for the graph property "the graph can be expressed as a union of triangles", right?  (ii) By union, do you mean disjoint union as in your example?  Or just regular union?
Neal Young
CC BY-SA 4.0
2021-10-27T11:58:19.330Z
Your example shows that a vertex could belong to two (or more?) triangles, but it is not clear if you are allowing edges to belong to two (or more) triangles as well. Since you ask for a name OR a property, perhaps a characterizing property would be "a graph where every vertex and every edge is in a triangle".
JimN
CC BY-SA 4.0
2021-10-27T12:08:22.787Z
As a bigger picture, if you are just dealing with triplets and trying to analyze how they interact with each other, then consider the possibility that a graph data structure is *not* an ideal way to look at them. You should probably be looking at sets-of-3set structures, or block designs. For example, if you have triangles abd, bce, acf and you form your graph, then the graph would have four triangles (abd, bce, acf and also triangle abc) while your initial set of triplets was three 3sets.
JimN
<graph-theory><ds.data-structures>
Does a graph resulting from the union of triangles has a particular name?
jesús garcía
# Does a graph resulting from the union of triangles has a particular name?

(Asked by: username_0 on 2021-10-14)

I have different simple triangle graphs. As an instance, $G\_1=(V\_1,E\_1)=(\{1,2,3\},\{\{1,2\},\{2,3\},\{3,1\}\})$ and $G\_2=(V\_2,E\_2)=(\{1,4,5\},\{\{1,4\},\{4,5\},\{5,1\}\})$.

The union of both graphs is $H=(V\_1 \cup V\_2, E\_1 \cup E\_2)=(\{1,2,3,4,5\},\{\{1,2\},\{2,3\},\{3,1\},\{1,4\},\{4,5\},\{5,1\}\})$

Does a graph like $H$ that results from the union of triangles have a particular name or properties?

**Tags:** graph-theory, ds.data-structures
> **Comments:**
> **username_1**: (i) You are looking for a name for the graph property "the graph can be expressed as a union of triangles", right?  (ii) By union, do you mean disjoint union as in your example?  Or just regular union?  
> **username_2**: Your example shows that a vertex could belong to two (or more?) triangles, but it is not clear if you are allowing edges to belong to two (or more) triangles as well. Since you ask for a name OR a property, perhaps a characterizing property would be "a graph where every vertex and every edge is in a triangle".  
> **username_2**: As a bigger picture, if you are just dealing with triplets and trying to analyze how they interact with each other, then consider the possibility that a graph data structure is *not* an ideal way to look at them. You should probably be looking at sets-of-3set structures, or block designs. For example, if you have triangles abd, bce, acf and you form your graph, then the graph would have four triangles (abd, bce, acf and also triangle abc) while your initial set of triplets was three 3sets.  

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## 1 answers

### Answer by username_3 on 2021-10-22. Score: 1

According to [Wikipedia](https://en.wikipedia.org/wiki/Butterfly_graph), this graph is known variously as the **butterfly graph**, **bowtie graph**, or **hourglass graph**.

> **Comments:**
> **username_1**: With the caveat that the butterfly/bowtie/hourglass graph is just one particular graph.  It is not a general name for any graph that can be formed as the union of triangles.  
> **username_3**: Ah, indeed. I was only looking at the graph $H$ from the question. I think the OP would need to provide some more examples where there are >2 triangles in the union in order to see more clearly what family of graphs are under consideration (E.g.: are all of the triangles joined only at one common vertex or are there other ways of joining them?)  

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