Draw edge on arc

Question

I would like to draw an edge between two nodes which lies on a clip of an circle. For small angles I was able to get some more or less realistic appearance by using bend left and bend right, but for large distances, for example to connect a node which lies at 90° with an node at 225°, this is getting ugly.

However, drawing an arc alone does not draw arrows and does not end at the "borders" of the nodes, but at their center. So is there any solution, e.g. an argument for "edge" while building a path, where I can specify an arc on which it should lie?

Here is an example:

\documentclass{scrartcl}
\usepackage{tikz}
\begin{document}

\begin{tikzpicture}
\node (V1) at (0,8) {$\o{v}_1$};
\node (V3) at (4,4) {$\o{v}_2$};
\node (V5) at (0,0) {$\o{v}_3$};
\node (V6) at (-2.83,1.17) {$\o{v}_4$};

\node at (0,4) {$\o{C}$};

\path   (V1) edge[bend left=45] node[above right] {$\o{g}_1(x)$} (V3)
        (V5) edge[bend right=45] node[below right] {$\o{g}_2(x)$} (V3)
        (V5) edge[bend left=15] node[below] {$\o{g}_3(x)$} (V6)
        (V1) edge[bend right=70] node[above left] {$\o{g}_4(x)$} (V6);
\end{tikzpicture}
\end{document}

While the first three edges of the path look more or less as if the would lie on a circle, the fourth one makes the circle a potato.

Answer 1

you say:

drawing an arc alone does not draw arrows and does not end at the "borders" of the nodes, but at there center

(*their center, b.t.w.)

but it is not completely true; you can try with a workaround with arc. For example:

\draw [right, ->] (0.3,8) arc [radius=4, start angle=90, end angle= 0];

will draw an arrow from V1 to almost V2.

Playing around with the radius and the location of the nodes you can draw your circle (with arrows, see the -> above)

EDIT:

to avoid playing around with the starting location you can use node anchors:

\draw [->] (V1.east) arc [radius=4, start angle=90, end angle= 0];

Answer 2

Another arc solution:

\documentclass{scrartcl}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}

\begin{tikzpicture}
\node (V1) at (0,8) {$\o{v}_1$};
\node (V3) at (4,4) {$\o{v}_2$};
\node (V5) at (0,0) {$\o{v}_3$};
\node (V6) at (-2.83,1.17) {$\o{v}_4$};

\node at (0,4) {$\o{C}$};

\draw[-latex,shorten <= 10pt,shorten >= 10pt] (V1) arc(90:0:4cm) node[above right=3ex and .5ex] {$\o{g}_1(x)$} (V3);
\draw[-latex,shorten <= 10pt,shorten >= 10pt] (V5) arc(-90:0:4cm) node[below right=3ex and .5ex] {$\o{g}_2(x)$} (V3);
\draw[-latex,shorten <= 10pt,shorten >= 10pt] (V5) arc(270:225:4cm) node[below right=3ex and -2.5ex] {$\o{g}_3(x)$} (V6);
\draw[-latex,shorten <= 10pt,shorten >= 10pt]        (V1) arc(90:225:4cm) node[above left =1ex and 2.5ex] {$\o{g}_4(x)$} (V6);
\end{tikzpicture}
\end{document}

Adjust the start angle, end angle and shorten (both <= and >=) distances as per taste.