given six real-analytic arcs in the unit disk $D$, each of which connects the origin to a boundary point, and no two arcs meet anywhere except at the origin, and the arcs meet at equal (60 degree) angles at the origin, is there a homeomorphism $\phi$ of $D$ that takes the six arcs onto six straight line segments connecting the origin to equally spaced points on the boundary (like a sliced pizza, hence the name "pizza lemma")? And if so can $\phi$ be made $C^2$ or even smoother?

I can get it $C^2 $if I assume the original arcs are (all the) zeroes of some harmonic function. I can get it $C^0$ with no additional assumption if I am willing to use the Riemann mapping theorem (on each sector and then patch up the sector boundaries). But I am hoping somebody knows a theorem of which this is an immediate application so I can just quote it.

Similarly, the "cake lemma", if there are three arcs that do not meet connecting six boundary points in pairs, there should be a homeomorphism of $D$ that takes the three arcs onto three parallel straight lines. As the first commenter pointed out, that's false without an additional assumption, so assume that the points are 0,1,2,3,4,5, and points 0,1,2 are each one endpoint of one of the three arcs.

More generally, given a homeomorphism between two 1-complexes that are subsets of $D$, when does there exist a homeomorphism of $D$ that restricts to the given homeomorphism of subsets?