What may be the definition of the shortest smooth continuously differentiable plane curve through four given points including three given self-intersections among them?
Thanks for exploration, am new to graph theory; hope at least the question makes sense, so included a sketch of the $ 4+3=7$ points.
Some path consisting of straight lines is the shortest. But it’s not “smooth”. You can make it smooth by adding little fillets at the sharp corners. By making the fillets very small, you can get an arclength that’s arbitrarily close to the straight-line path.
If you want something even smoother, consider this example. Take the points $A=(-1,0)$, $B=(0,1)$, $C=(1,0)$. The shortest path from $A$ to $C$ obviously consists of two straight lines. But you could also draw a hyperbola passing through the three points, with it’s “peak” (point of maximum curvature) at $B$. By making the peak point sharper, you can get arbitrarily close to the straight line solution. In fact, in the limit, the hyperbola becomes a pair of straight lines.