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Named after the tendency of cats to bury their excreta, a cat hole is a small hole1) dug to contain solid waste. It is dug away from use areas and water sources, and is covered after a single use.
Leaving toilet paper in the cat hole is a matter of local policy:
Check with the local BLM/NF website or collateral materials for guidance on proper cat-holing procedure in the area.
The use of a narrow "trench spade" shovel will be easier than a broadheaded shovel in hard earth conditions. Plastic camping trowels are light weight, smaller in size than a spade and work well in soft soils. Metal garden trowels, while heavier, will stand up better to rocky or hard soils.
Note: secessus has begun using/recomending a mattock for cat hole digging duties. His was $10 at at flea market.