<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e276" altimg="si32.svg"><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-colourings and Ferrers diagram representations of cographs

For a pair of natural numbers k,l, (k,l)-colouring graph G is partition the vertex set into (possibly empty) sets S1,S2,…,Sk, C1,C2,…,Cl such that each Si an independent and Cj induces clique in G. The problem, which NP-complete general, has been studied for special classes as chordal graphs, cographs line graphs. Let ?ˆ(G)=(?0(G),?1(G),…,??(G)?1(G)) ?ˆ(G)=(?0(G),?1(G),…,??(G)?1(G)) where ?l(G) (respectively, ?k(G)) minimum k l) (k,l)-colouring. We prove ?ˆ(G) ?ˆ(G) are conjugate sequences every when cograph, number vertices equal to sum entries or ?ˆ(G). Using decomposition property we show cograph can be represented by Ferrers diagram. devise algorithms compute find induced subgraph used certify non-(k,l)-colourability