How many edges can a unipathic graph have?

Question

A unipathic graph is a directed graph such that there is at most one simple path from any one vertex to any other vertex.

Unipathic graphs can have cycles. For example, a doubly linked list (not a circular one!) is a unipathic graph; if the list has $n$ elements, the graph has $n-1$ cycles of length 2, for a total of $2(n-1)$.

What is the maximum number of edges in a unipathic graph with $n$ vertices? An asymptotic bound would do (e.g. $O(n)$ or $\Theta(n^2)$).

_{Inspired by Find shortest paths in a weighed unipathic graph; in my proof, I initially wanted to claim that the number of edges was $O(n)$ but then realized that bounding the number of cycles was sufficient.}

Answer 1

A unipathic graph can have $\Theta(n^2)$ edges. There's a well-known kind of graph that's unipathic and has $n^2/4$ edges.

Consider a complete bipartite graph, with oriented edges $\forall (i,j) \in [1,m]^2, a_i \rightarrow b_j$. This graph is unipathic and has no cycle: all its paths have length $1$. It has $2m$ vertices and $m^2$ edges.

_{(Follow-up question: is this ratio maximal? Probably not, but I don't have another example. This example is maximal in the sense that any one edge that you add between existing nodes will break the unipathic property.)}

Answer 2

I don't know if there a unipathic graph on more than $\frac{n^2}{4}$ edges, but here's an argument showing that no more than $\frac{n^2}{2}+3$ edges are possible:

Assume by contradiction that $G=(V,E)$ is a unipathic graph such that $|E|\geq \frac{n^2}{2}+3$.

By the pigeonhole principle, there exist $v\in V$ such that $$d_{\text{in}}(v)\geq \frac{n}{2}+1$$

Denote $U=\{u\in V\mid (u,v)\in E\}$

Notice that if there was a vertex $x\in V\setminus \{v\}$ such that $$\exists u_1\neq u_2\in U:(x,u_1),(x,u_2)\in E$$

Then the graph wouldn't be unipathic (as $(x\to u_1\to v)$ and $(x\to u_2\to v)$ are both valid paths).

This means that (adding the edges from $\{v\}\times U$): $$|E\cap (V\times U)|\leq2|U|$$

So the average in-degree of $U$'s vertices is at most 2, hence: $$|E|=|E\cap (V\times U)|+|E\cap (V\times (V\setminus U))|$$ $$\leq 2|U|+n|V\setminus U|\leq 2(\frac{n}{2}+1)+n(\frac{n}{2}-1)<\frac{n^2}{2}+3$$

$\square$