There is given a construction of ordered k-partite digraph $G=(V_{1},...,V_{k};E)$ :
- for every $v \in V$ $d_{in}(v)\ge1$ (in-degree)
- for every $v \in V_{1}$ $d_{out}(v)=0$ (out-degree)
- for $m>l$ $v_{1},v_{2} \in V_{l}$ and $v_{3} \in V_{m}$ is not allowed to $v_{1}v_{3} \in E$ and $v_{2}v_{3} \in E$ at the same time.
- $v \in V_{l}; v_{1} \in V_{m}; v_{2} \in V_{n}$ it is forbidden to have $vv_{1},vv_{2} \in E$, where $l>m,n$
- if $v_{1},v_{2} \in V_{a}$, $v_{3} \in V_{b}$, $v_{4}\in V_{c}$, $v_{5}\in V_{d}$ and $v_{1}v_{4},v_{2}v_{5} \in E$ then we can't take $v_{3}v_{1}, v_{3}v_{2} \in E$ where $a < b \le c,d$
- for some $v,u \in V$ if $Star_{in}(v)=Star_{in}(u)$ then it is improper to have $|Star_{in}(v)|=1$
Does a graph with that given properties exist? If so, please give a example of it and some other properties. I am asking because i do some research on k-partite digraphs and i have added as many conditions as it it possible
Regards.