<p>According to the 59th slide of the following pdf:</p>

<p><a href="https://grow2015.sciencesconf.org/file/174789" rel="nofollow noreferrer">https://grow2015.sciencesconf.org/file/174789</a> (A talk by D.Kratsch at GROW 2015)</p>

<p>we have that </p>

<blockquote>
  <p>A vertex in a graph is weakly simplicial if its neighborhood is an
  independent set and the neighborhoods of its neighbors form a
  chain under inclusion.</p>
</blockquote>

<p>See the slide for an illustration. The context where this definition is found in the slides is chordal bipartite graphs, so I guess this might be what you are looking for.</p>

CC BY-SA 3.0
2017-01-18T13:42:43.550Z
Observe that the hypergraph defined by a chordal bipartite graph is what is called a $\beta$-hypergraph. In this case then, taking the definition found by @a-sh, a weakly simplicial vertex $x$ is also called nested point or $\beta$-leaf (the edges of the hypergraph containing $x$ are ordered by inclusion) : see https://arxiv.org/abs/1403.7076. Such vertices are used to characterized $\beta$-acyclicity of hypergraphs or chordality of bipartite graphs: a bipartite graph $(X,Y,E)$ is chordal iff you can iteratively remove weakly simplicial vertices from $X$, until $X$ becomes empty.
holf
CC BY-SA 3.0
2017-01-22T09:47:06.247Z
@A.Sh Thank you very much. It is what I am looking for.
Gunelle
CC BY-SA 3.0
2017-01-22T09:48:02.287Z
@holf Thank you, too, for giving another view of weakly simplicial.
Gunelle
CC BY-SA 3.0
2017-01-22T10:17:12.950Z
@A.Sh However, does chain under inclusion mean the chain is formed by the neighborhood of the weakly simplicial vertex and the neighborhood of its neighbors?
Gunelle
CC BY-SA 3.0
2017-01-22T10:39:48.913Z
As far as I can parse the definition, let $u$ be a weakly simplicial vertex. Then the components of the chain are the neighborhoods of neighbors of $u$, i.e. if $\{v_i\}$ is the neighborhood of $u$, and $V_i$ denote the neighborhood of each $v_i$, then the set $\{V_i\}$ (ordered appropriately) should be a chain under inclusion.
M.B.
CC BY-SA 3.0
2017-01-22T11:50:39.770Z
@A.Sh In this case, such a chain is not possible as its components are all part of one partite set. Have I missed something?
Gunelle
CC BY-SA 3.0
2017-01-22T12:00:42.803Z
I think that the top vertices in the figure in the slide are supposed to be weakly simplicial. Let's enumerate the top vertices $s_1,s_2$ etc. from left to right, and the bottom ones $f_1,f_2$ etc. from left to right. The neighborhood of, say, $s_1$, is $\{f_1,f_2,f_3\}$. Then the corresponding neighborhoods, let's call them $F_1$ through $F_3$, are $F_1=\{s_1,s_2\}$, $F_2=\{s_1,s_2,s_3\}$ and $F_3=\{s_1,s_2,s_3,s_4\}$. Clearly $F_1 \subset F_2 \subset F_3$, and so $s_1$ is weakly simplicial.
M.B.
CC BY-SA 3.0
2017-01-22T12:14:56.160Z
@A.Sh Now I understand. At first I thought chain meant a succession of vertices joined by edges. F1⊂F2⊂F3 makes sense. Thank you!
Gunelle
CC BY-SA 3.0
2017-01-18T00:03:29.867Z
2017-01-18T00:03:29.867Z
M.B.
<p>While studying chordal bipartite graphs, I have come across <strong><em>weakly simplicial</em></strong> vertices. I have searched for some time to understand what a weakly simplicial vertex is but I haven't succeeded. </p>

<p>A simplicial vertex is a vertex whose neighborhood induces a clique. </p>

<p>So, what is the weak form of this definition?  </p>

CC BY-SA 3.0
2017-01-17T16:35:26.540Z
Where did you see this?
holf
<graph-theory><bipartite-graphs>
What is a weakly-simplicial vertex?
Gunelle
# What is a weakly-simplicial vertex?

(Asked by: username_0 on 2017-01-17)

While studying chordal bipartite graphs, I have come across ***weakly simplicial*** vertices. I have searched for some time to understand what a weakly simplicial vertex is but I haven't succeeded.

A simplicial vertex is a vertex whose neighborhood induces a clique.

So, what is the weak form of this definition?

**Tags:** graph-theory, bipartite-graphs
> **Comments:**
> **username_1**: Where did you see this?  

---

## 1 answers

### Answer by username_2 on 2017-01-18. Score: 4 | Accepted answer

According to the 59th slide of the following pdf:

<https://grow2015.sciencesconf.org/file/174789> (A talk by D.Kratsch at GROW 2015)

we have that

> A vertex in a graph is weakly simplicial if its neighborhood is an
> independent set and the neighborhoods of its neighbors form a
> chain under inclusion.

See the slide for an illustration. The context where this definition is found in the slides is chordal bipartite graphs, so I guess this might be what you are looking for.

> **Comments:**
> **username_1**: Observe that the hypergraph defined by a chordal bipartite graph is what is called a $\beta$-hypergraph. In this case then, taking the definition found by @a-sh, a weakly simplicial vertex $x$ is also called nested point or $\beta$-leaf (the edges of the hypergraph containing $x$ are ordered by inclusion) : see https://arxiv.org/abs/1403.7076. Such vertices are used to characterized $\beta$-acyclicity of hypergraphs or chordality of bipartite graphs: a bipartite graph $(X,Y,E)$ is chordal iff you can iteratively remove weakly simplicial vertices from $X$, until $X$ becomes empty.  
> **username_0**: @A.Sh Thank you very much. It is what I am looking for.  
> **username_0**: @holf Thank you, too, for giving another view of weakly simplicial.  
> **username_0**: @A.Sh However, does chain under inclusion mean the chain is formed by the neighborhood of the weakly simplicial vertex and the neighborhood of its neighbors?  
> **username_2**: As far as I can parse the definition, let $u$ be a weakly simplicial vertex. Then the components of the chain are the neighborhoods of neighbors of $u$, i.e. if $\{v_i\}$ is the neighborhood of $u$, and $V_i$ denote the neighborhood of each $v_i$, then the set $\{V_i\}$ (ordered appropriately) should be a chain under inclusion.  
> **username_0**: @A.Sh In this case, such a chain is not possible as its components are all part of one partite set. Have I missed something?  
> **username_2**: I think that the top vertices in the figure in the slide are supposed to be weakly simplicial. Let's enumerate the top vertices $s_1,s_2$ etc. from left to right, and the bottom ones $f_1,f_2$ etc. from left to right. The neighborhood of, say, $s_1$, is $\{f_1,f_2,f_3\}$. Then the corresponding neighborhoods, let's call them $F_1$ through $F_3$, are $F_1=\{s_1,s_2\}$, $F_2=\{s_1,s_2,s_3\}$ and $F_3=\{s_1,s_2,s_3,s_4\}$. Clearly $F_1 \subset F_2 \subset F_3$, and so $s_1$ is weakly simplicial.  
> **username_0**: @A.Sh Now I understand. At first I thought chain meant a succession of vertices joined by edges. F1⊂F2⊂F3 makes sense. Thank you!  

---

