Soft question: meaning of 'disjoint' (i.e. is it edge or vertex disjoint?) in definition of a fan.

Question

I am working throuhg some questions of graph theory, and in one of the question I am given the definition:

Let $B\subset V(G)$ and $a$ be a vertex not in $B$. An $a$-$B$ fan is a family of $|B|$ paths from $a$ to $B$, disjoint except at $a$.

I am not sure if by 'disjoint' we mean edge-disjoint, or the stronger condition 'vertex disjoint' (clearly if it is vertex disjoint, it must be edge-disjoint). I have not come across the definition of an $a$-$B$ graph online, only a fan-graph $F_{n,m}$ which is something different.

Answer 1

There are a handful of cues which all point towards the “vertex-disjoint” interpretation:

1) If it were merely edge-disjoint, there would be no (immediate) need to specify “except at $a$”. In fact the expression “edge-disjoint except at $a$” is already suspect: if two paths are not edge-disjoint at $a$ then it’s because they share an additional vertex besides $a$. So it seems odd to call this “except at $a$” rather than “except at edges incident to $a$”.

2) If it were only edge-disjoint, there would be no special significance to having $|B|$ paths, since they wouldn’t be forced to cover all of $B$.

3) Many terms in graph theory tend to be vertex-oriented by default if not otherwise specified: chromatic number vs edge-chromatic number; $k$-connected vs $k$-edge-connected. And of course, the size of a graph $|G|$ is customarily the number of vertices, not edges.

Taken together, the evidence is strong that edge-disjoint is not the intent. If we relaxed the definition to merely edge-disjoint, then it might allow an $(a,B)$-fan (where $|B|=3$ and $b \in B$ is adjacent to $a$) to consist of 3 copies of the exact same edge from $a$ to $b$. After all, those paths are edge-disjoint allowing for the exception at $a$ (yes they also meet at $b$, but see discussion at point 1) and there are 3 of them.

It’s hard to believe someone would want to define the term “fan” to include such an outcome.

Answer 2

In West's Introduction to Graph Theory, the definition is

Given a vertex $x$ and a set $U$ of vertices, an $x,U$-fan is a set of paths from $x$ to $U$ such that any two of them only share the vertex $x$.

This leaves out the condition about having exactly $|U|$ paths, but later quotes Dirac's Fan Lemma as saying

A graph is $k$-connected if and only if it has at least $k+1$ vertices and, for every choices of $x,U$ with $|U|\ge k$, it has an $x,U$-fan of size $k$.

In the most interesting case, we take $|U|=k$, and then the number of paths is $|U|$ as well.