For some reason, I saw, or read this question as simply asking "How fast can you get to Mars if fuel is no expense". So this answers that question. I don't mind deleting it if that is considered appropriate.
Anyway, it depends what you mean by "if fuel is no expense". If you are assuming something more or less along the lines of current rockets, but are willing to use more fuel, and so carry less payload than is usually considered, there are trajectories that get you to Mars in three or four months within the capabilities of rockets currently being designed. The don't look very different from your picture, except that the transfer orbit actually reaches out past the orbit of Mars, although the rocket doesn't do that part of the orbit because it arrives at Mars first. Chemical rockets have nowhere near the capability to counter Earth's orbital motion (30 km/s roughly) and go in "the opposite direction" for instance. The usual way to represent the options for this kind of interplanetary trajectory is called a "pork chop plot" and shows options in terms of departure date, arrival date (or journey length) and delta-V required. There are numerous answers on this stack exchange which include examples of them.
If on the other hand, you imagine a "magical" rocket, powered by fusion or antimatter which can sustain a 1g acceleration continuously (much more would be uncomfortable for the crew) then the fastest trajectory looks very much like a straight line, and takes about 40 hours at closest approach (just use $s = \frac{1}{2} a t^2$ separately on each half of the journey. The motions of the planets and the Sun's gravity will be a small correction to this.
Finally if you just want to ship very small robust cargo items (such as individual protons or neutrinos) you could send them at close to the speed of light, for an absolute minimum delivery time of about 7 minutes.