There are a zillion different types of sedum so it depends on what you choose....some species will work better than others. I'd focus on the evergreen, ground covering stonecrop types - Evergreen Sedum Stonecrop.

Sun is the preferred growing condition and those in more shade will be sparser and likely not bloom well. Also, these tend to produce white or yellow flowers, very rarely in pinks, reds or purples. You might also want to consider creeping phlox or thyme.

If you have any kind of garden at all, it is almost impossible to avoid toxic plants. There are far more common and popular garden plants that are toxic than those that are not. But dogs and cats tend not to be attracted to these plants outdoors nor want to eat them and cases of pet poisonings are extremely rare. I:ve gardened for decades with an assortment of dogs and cats and a garden full of potentially toxic plants with zero issues!! This is really a needless concern.

The landscaping company that installed your paver patio should have installed rigid edging surrounding it. This is SOP for a paver or flagstone patio and would eliminate any erosion or sand migration.

Where are you finding 2 ft tall liriope? It sounds like something I need. Mine is about 6 inches tall. The leaves are longer, maybe 9 inches, but they flop over. Personally, I'd give serious thought to using it in this application. It can handle light foot traffic, and doesn't need an edging between it and the lawn. The border is just where the mower runs.

From my reading, it sounds like the liriope leaves grow to about a foot but the flower adds about another foot. That said, I haven't counted it out!

It's one thing to have somewhat toxic plants in like a garden bed, but I'd like to avoid it where the dog will step all the time. Actually, to this point, I have all kinds of plants in the yard and almost all are edible for both dogs and humans. Deliberately so. Certainly in the backyard.

I think the phlox or thyme is maybe too short. Since I need the plant to stay taller than the bricks in the event of some sinking.

Another plant that sounds like it "might" work for my border, on paper, is sea thrift. But I figured if it worked so well in my area, I'd see it planted more often.

I've found all kinds of beautiful SunSparkler sedums on Proven Winners website with pink or red flowers. Again, "on paper" sounds like it would work. Just don't want to destroy my pavers with those roots!

The pavers have like this plastic edging nailed into the ground. It winds up getting exposed over time as the sand smothers the grass at the edge. Then the ground there starts to sink and you can see the plastic, which is supposed to be covered by soil and grass, but the ground continually erodes around it. So every summer is a cycle of adding another layer of soil and resodding. Obviously there is some kind of grading issue, but truly remedying that seems like a herculean undertaking. At this point it is easier to just cover up the problem with a hardier plant.

Liriope isn't nearly that big. It's just another illiterate plant .

9 inches in bloom

Can you post one or two pictures of your patio edge? I don't think there are any sedums in zone 5 whose roots would upset your pavers. If you used a small one like Sedum Acre or Dragon's Blood, they might spread between the pavers, but it sounds to me like you want something a little bigger around the edge. If grass is dying at the edge, that makes me wonder if it's suffering from reflected heat and lack of water. A larger sedum like Autumn Joy might look nice. I'm not familiar with Sunsparkler sedums - - I'll have to look that up.

I have yet to see any of the Sunsparkler series to be very robust growers nor remain fully evergreen, even in my mild winter climate. I have been growing 'Lime Zinger' - supposedly "evergreen in mild climates" for a number of years and it dies back completely to the ground in winter.

Just don't want to destroy my pavers with those roots!

Sedum roots won't destroy anything.

This is mostly a picture of my calamansi last summer, but you can see my problem with the exposed edging. I don't think the problem is reflected light or any of that; I think it is a slight slope encouraging runoff from the patio to roll downhill, smothering the grass on its way.

It doesn't look this bad right now because we added more soil and grass, but by this fall it may well look this way again.

Not much stays evergreen here, so I am not expecting that. Just hoping for it to come back each spring.

I am glad to hear the sedum roots aren't destructive.

Remediation attempts by the patio company are always temporary fixes.

==>>>

what????

most likely your failure to grow the greatest weed on the face of the earth.. grass.. is lack of water next to bricks that become blistering hot in summer.. and hold that heat deep into the night.. so the grass does not have its usual night recovery time..

grass will thrive in the cracks of cement and blacktop.. failure of grass to grow is contrary to nature.. lol ..

sooo.. if watering is the problem.. no matter what you plant.. you will have problems with it ...

the alternative.. is if they used a very out of wack pH crushed stone under the brick ....

all that said.. i used sedum in walkways.. and it took all the punishment i could give it.. and it seeded into the cracks of the blacktop... and its rather drought resistant.. but you will have to water it the first year.. to get it fuilly established.. which mean.. grow the root mass requisite to take all the abuse ...

i check later to see if i have pix.. i used the old dragon blood red flowered.. and its twin yellow flowered one.. as well as the rouge sedum that lived in my lawn which was former horse pasture.. sorry dont know the latin on any of them ...

really.. remediation.. this isnt a brown field site ...

ken

ps: who installed this.. a bigbox handyman... or a nurseryman who dabbles in brick.. one one think a plantsman .. would have better answers to why this famous weed wont grow ...

btw.. imo .... you should have added the perennial forum ... and skipped one of the other forums you listed..

Something doesn't seem to add up to me, at least with this pic and the description. IMO, unless you are trying to have landscape beds to fully enclose your patio, having a perimeter of plantings to step over kinda defeats the purpose of the patio opening up to the yard. And if you can't even get grass to grow, . . . .

- The edge pavers should not be sinking. The paver gravel base should extend out beyond the paver perimeter a good 6". The paver plastic edging leaves about 1" exposed of the paver height, enough to bring sod close.

- It looks like polymeric sand in your joints, if so that would not wash away. However, maybe upon first application, or re-apps, the "overthrow" not in the joints that was washed off into the (about 12" wide?) perimeter soil just hasn't recovered. And the topping off with more soil doesn't look like it was a good nutrient rich blend.

- Water running off should not kill the sod, unless there is high traffic in the winter time on the soggy dormant sod.

- Because of the erosion, I suspect there is a slope up against the perimeter that is steeper than the general yard (about 12" wide?), and the mower is not running parallel, or too wide if done so, and it is lopping the grass too short leading to brownout and dieoff.

It was definitely sand washing out as we could see the sand. Patio company came back many times to refill. It was a dedicated hardscaping company, not big box store. I think they don't know how to deal with slopes since almost everywhere is completely flat around here, except our yard apparently.

Sod is brought right beside the edging, but end of season winds up being this way. Almost 5 years going.

I didn't say the edge pavers were sinking, just the ground around it. So bricks stay high while ground gets low, exposing the plastic edging.

Having the plant perimeter (not grass) will also prevent the lawnmower from needing to get close to the patio, to also address that point. And also don't need to worry about the landscaper hitting the bricks, which has also happened previously.

I don't mind the thought of low growing plants "enclosing" the patio. I think it could even look nice.