Drawing a circular graph using xypic

My it is only a comment instead of a true answer, because I think that with xymatrix the way is very very long and with lot of difficulties.

documentclass[a4paper,12pt]{article}
usepackage[all]{xy}
usepackage{newtxtext,newtxmath}

begin{document}
xymatrix{
G& {overset{displaystyle 1}{bullet}}ar@/^1pc/[dr]^{alpha_1} &  
A& B& {bullet}ar@/^1pc/[dl]
D& E& F}
end{document}

My problems are:

1. @{.} give me only dotted straight line and not curved arrows withouts tips. In addendum in the picture there are only 3 dots.
2. To put the rotated circle labels. For 1 it is very simple but for the others (????).
3. Create a fast circle loop (see the § 8.2 Circles and Ellipses of the complete/old documentation http://texdoc.net/texmf-dist/doc/generic/xypic/xyrefer.pdf).

I should read and to understood the manual and the commands that they are differents from the canonical but not very simple use of the arrows.

Compile with Asymptote

unitsize(3cm);

pair[] A;
for (int i=0; i<8; ++i){ A[i]=dir(90-i*45); }

arrowbar Ar=Arrow(TeXHead,2);
margin Tr=TrueMargin(10,10);
margin Tr1=TrueMargin(10,15);
margin Tr2=TrueMargin(15,10);

dot("$1$",A[0],2*dir(A[0]));
dot("$2$",A[1],2*dir(A[1]));
path a1=arc(0,A[0],A[1],false);
draw(Label("$alpha_1$",2*LeftSide),a1,Ar,Tr);
dot("$3$",A[2],2*dir(A[2]));
path a2=arc(0,A[1],A[2],false);
draw(Label("$alpha_2$",2*LeftSide),a2,Ar,Tr);
dot("$i$",A[4],2*dir(A[4]));

frame f;
draw(f,(-7.5,0)--(7.5,0),dotted+2bp);
add(rotate(degrees(dir(A[3]))+90)*f,A[3]);
add(rotate(degrees(dir(A[5]))+90)*f,A[5]);

draw(Label("$alpha_3$",2*LeftSide),arc(0,A[2],A[3],false),Ar,Tr1);
draw(Label("$alpha_{i-1}$",2*LeftSide),arc(0,A[3],A[4],false),Ar,Tr2);
draw(Label("$alpha_i$",2*LeftSide),arc(0,A[4],A[5],false),Ar,Tr1);
draw(Label("$alpha_{n-2}$",2*LeftSide),arc(0,A[5],A[6],false),Ar,Tr2);

dot("$n-1$",A[6],2*dir(A[6]));
dot("$n$",A[7],2*dir(A[7]));
draw(Label("$alpha_n$",2*LeftSide),arc(0,A[7],A[0],false),Ar,Tr);
draw(Label("$alpha_{n-1}$",2*LeftSide),arc(0,A[6],A[7],false),Ar,Tr);