How to draw a digraph without intersecting arcs

Question

I am tryng to draw the following figure that is a graph with bipartition $X$ and $Y$:

I'm having trouble drawing arcs so they don't intersect. Follows the my code that alreadys did:

 \begin{tikzpicture}[
 node distance = 5mm and 33mm,
 start chain = going below,
 doc/.style = {
 dot,
 on chain
 },
 dot/.style = {circle, fill, inner sep=0pt, outer sep=0pt, minimum size=3mm,
 node contents={}},
 FIT/.style = {ellipse, draw, thick, inner xsep=2em, yshift=-1ex, fit=#1},
 ->-/.style = {decoration={markings,
 mark=at position .5 with {\arrow{Straight Barb}}},          draw, postaction={decorate}             },
 every edge/.append style = {very thick, dotted, shorten <=3mm, shorten >=3mm}
 ]
 % vertices
 \foreach \i [count=\j] in {1,2,3,4,5,6,t-1,t}%
 {
 \ifnum\j=6 
 \node (m\j) [doc,fill=none];
 \else 
 \node (m\j) [doc,label=left:{$a_{\i}$}];
 \node (n\j) [dot, right=of m\j, label=right:{$b_{\i}$}];
 \draw[->-, red] (m\j) -- (n\j);
 \fi
 }

 \draw[->-, blue]
 (m2) -- (n1);
 \draw[->-, blue]
 (m1) -- (n2);

 \draw[->-, blue]
 (n4) -- (m5);
 \draw[->-, blue]
 (n3) -- (m4);

 \draw[->-, blue]
 (n3) -- (m4);

 \draw[->-, blue]
 (n7) -- (m8);

 % dashed line
 \draw   (m5) edge (m7)  (n5) edge (n7);

 \node[FIT=(m1) (m7), label=$X$] {};
 \node[FIT=(n1) (n7), label=$Y$] {};
 \end{tikzpicture}

For example in my code the arcs $(b_1,a_2)$ and $(b_2,a_1)$ intersects. I wish to draw as shown in the figure. How I can to do?

Answer 1

Like this?

\documentclass[margin=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,
                chains,
                decorations.markings,
                fit,
                positioning,
                shapes.geometric,
                }

\begin{document}
    \begin{tikzpicture}[
 node distance = 5mm and 33mm,
   start chain = going below,
    doc/.style = {dot, on chain},
    dot/.style = {circle, fill, 
                  inner sep=0pt, outer sep=0pt, minimum size=3mm,
                  node contents={}},
    FIT/.style = {ellipse, draw, thick, inner xsep=2em, yshift=-1ex, fit=#1},
    ->-/.style = {decoration={markings,
                  mark=at position .5 with {\arrow{Straight Barb}}},
                  draw=#1, postaction={decorate}             },
    ->-/.default = red,
every edge/.append style = {very thick, dotted, shorten <=3mm, shorten >=3mm}
                        ]
% vertices
\foreach \i [count=\j] in {1,2,3,4,5,6,t-1,t}%
{
\ifnum\j=6
\node (m\j) [doc,fill=none];
\else
\node (m\j) [doc,label=left:{$a_{\i}$}];
\node (n\j) [dot, right=of m\j, label=right:{$b_{\i}$}];
\draw[->-] (m\j) -- (n\j);
\fi
}
 %
    \begin{scope}[->-/.append style=blue],
\draw[->-]  (n1) -- (m2);
\draw[->-, semithick, rounded corners=2mm]   
            (n2.45) -- ([xshift=7mm] n1.south) 
                    |- ([yshift=2mm] n1.north) 
                    -| (m1);
\draw[->-]  (n4) -- (m5);
\draw[->-]  (n3) -- (m4);
\draw[->-, semithick, rounded corners=2mm]
            (n5.45) -- ([xshift=7mm] n4.south)
                    |- ([yshift=2mm] n3.north)
                    -| (m3);
\draw[->-]  (n7) -- (m8);
\draw[->-, semithick, rounded corners=2mm]
            (n8.45) -- ([xshift=11mm] n7.south)
                    |- ([yshift=2mm] n7.north)
                    -| (m7);
    \end{scope}

% dashed line
\draw   (m5) edge (m7)  (n5) edge (n7);

\node[FIT=(m1) (m7), label=$X$] {};
\node[FIT=(n1) (n7), label=$Y$] {};
    \end{tikzpicture}
\end{document}