Partition of a Binary Matrix into 𝑘 (𝑘≥3) Exclusive Row and Column Submatrices Is Difficult

Table 1

The relationship between the clause 𝑐_𝑖={𝑢_𝑥,𝑢_𝑦,𝑢_𝑧} and the vertex values of 𝐵_𝑖.

(𝐴(𝑢𝑥), 𝐴(𝑢𝑦), 𝐴(𝑢𝑧))	(𝑇, 𝐹, 𝐹)	(𝐹, 𝑇, 𝐹)	(𝐹, 𝐹, 𝑇)
3-biclique partitions of 𝑇𝑖1	{𝑙0,𝑟0,𝑞𝑖1,𝑞𝑖2,𝑞𝑖4} /{𝑙1,𝑟1,𝑢𝑥,𝑑𝑖3} /{𝑙2,𝑟2,𝑑𝑖2,𝑞𝑖3}	{𝑙0,𝑟0,𝑢𝑥,𝑑𝑖2} /{𝑙1,𝑟1,𝑑𝑖3,𝑞𝑖1} /{𝑙2,𝑟2,𝑞𝑖2,𝑞𝑖3,𝑞𝑖4}	{𝑙0,𝑟0,𝑢𝑥,𝑞𝑖2} /{𝑙1,𝑟1,𝑞𝑖1,𝑞𝑖3,𝑞𝑖4} /{𝑙2,𝑟2,𝑑𝑖2,𝑑𝑖3}
3-biclique partitions of 𝑇𝑖2	{𝑙0,𝑟0,𝑢𝑦,𝑛𝑖3} /{𝑙1,𝑟1,𝑛𝑖1,𝑛𝑖2} /{𝑙2,𝑟2,𝑑𝑖2,𝑚𝑖1}	{𝑙0,𝑟0,𝑑𝑖2,𝑛𝑖1} /{𝑙1,𝑟1,𝑢𝑦,𝑚𝑖1} /{𝑙2,𝑟2,𝑛𝑖2,𝑛𝑖3}	{𝑙0,𝑟0,𝑢𝑦,𝑛𝑖3} /{𝑙1,𝑟1,𝑛𝑖1,𝑛𝑖2} /{𝑙2,𝑟2,𝑑𝑖2,𝑚𝑖1}
3-biclique partitions of 𝑇𝑖3	{𝑙0,𝑟0,𝑢𝑧,𝑜𝑖1} /{𝑙1,𝑟1,𝑑𝑖3,𝑝𝑖1} /{𝑙2,𝑟2,𝑝𝑖2,𝑝𝑖3}	{𝑙0,𝑟0,𝑢𝑧,𝑜𝑖1} /{𝑙1,𝑟1,𝑑𝑖3,𝑝𝑖1} /{𝑙2,𝑟2,𝑝𝑖2,𝑝𝑖3}	{𝑙0,𝑟0,𝑝𝑖1,𝑝𝑖2} /{𝑙1,𝑟1,𝑢𝑧,𝑝𝑖3} /{𝑙2,𝑟2,𝑑𝑖3,𝑜𝑖1}