How to build a satellite network around Kerbin which guarantees full coverage?

With update 1.2, you need to make sure that probes always have a communication link to Kerbin. There is also an advanced difficulty setting where you don't have stations all over Kerbin but only have the KSC. With that option on, each probe requires either direct line of sight to the KSC or to a communication satellite which has a connection.

How many relay satellites in what kind of orbits around Kerbin do you need to ensure you always have a connection to the KSC? Let's ignore occlusion by Mun and Minmus for now and focus only on what it takes to not get blocked by Kerbin.

Solution 1:

For most practical applications an easy solution is to have a ring of three evenly spaced satellites in high (2500km-7000km) circular equatorial orbits.

The only important thing is that the orbital periods of all three be the same, as the larger the difference, the sooner they will get out of position. Otherwise this scheme is resistant to placement errors.

Details:

• Number of satellites - Three satellites are enough to form a complete ring around Kerbin. You probably could get away with two, but the placement would be very tricky.
• Spacing - Generally, so long as any satellite can see the other two, everything will work, but placing them evenly will leave more room for error from orbital period mismatch to accumulate.
• Orbital height - In theory, you could go as low as double (Kerbin radius + atmospheric height), or 740km altitude, but the placement would have to be perfectly at 120 degrees phase angle. A high orbit is easier to work with. On the other end, if you go over 8970km, you will get into the Mun's sphere of influence, and your satellite will get thrown off trajectory when an encounter happens.
• Circular orbit - Just easier to work with (and to do the math).
• Equatorial - This system has blind spots on the surface. Placing the satellites in a roughly equatorial orbit leaves the blind spots at the poles (our main concern is that the KSC does not get into the blind spot).

Mun/Minmus occlusion: All the satellites can get occluded when either

• Kerbin, Mun and Minmus block one satellite each, or
• two satellites line up and both get occluded by Mun/Minmus, and the last one gets occluded by the other one.

This can be easily solved with the addition of one more satellite (at a later point), just make sure no two satellites line up when Kerbin blocks one of them.

(The "line up" will happen if, and only if, the phase angle differences between any one satellite, and any two other satellites adds up to somewhere near 180 degrees. So, for example, if we have three satellites 120 degrees apart, adding the fourth halfway between two of them is bad, as 120+60=180 and it can still get occluded. But adding it 40 degrees away is OK and will prevent occlusion from Mun/Minmus.)

Blind spots: Having a ring of satellites leaves blind spots on the surface (poles in case of equatorial ring). If the satellites are high up the spots will be really small, but any terrain irregularities close to them will easily block the signal. If you want probes at the poles, one solution is to just have another ring in a polar orbit. Or...

Rocket scientist alternative: if you are more of a rocket scientist than me, you can place four satellites in a tetrahedral formation around Kerbin. Such a formation will provide coverage on all of Kerbin's surface and cannot get fully occluded by Mun/Minmus. You can get more details in this KSP forum post, which takes it from this research paper (paywall).