Abstract is a forest? Can G be partitioned into a planar graph and a perfect graph? The NP-completeness of these problems are special cases of our result: if P and Q are additive induced-hereditary graph properties, then (P, Q)-colouring is NP-hard, with the sole exception of graph 2-colouring (the case where both P and Q are the set O of finite edgeless graphs). Moreover, (P, Q)-colouring is NP-complete iff P-and Q-recognition are both in NP. This completes the proof of a conjecture of Kratochvíl and Schiermeyer, various authors having already settled many sub-cases. Kratochvíl and Schiermeyer conjectured in [19] that for any additive hereditary graph properties P and Q, recognising graphs in P • Q is NP-hard, with the obvious exception of bipartite graphs (the case where both P and Q are the set O of finite edgeless graphs). They settled the case where Q = O, and it was natural to extend the conjecture to induced-hereditary properties. Berger's result [3] that reducible additive inducedhereditary properties have infinitely many minimal forbidden subgraphs provided support for the extended conjecture. We prove the extension of the Kratochvíl-Schiermeyer conjecture in this paper. Problems such as the following (for an arbitrary graph G) are therefore NP-complete. Can V (G) be partitioned into A ∪ B, so that G[A] is a line-graph and G [B] is a forest? Can G be partitioned into a planar graph and a perfect graph? For fixed k, , m, can G be partitioned into a k-degenerate subgraph, a subgraph of maximum degree , and an medge-colourable subgraph? Garey et al. [15, 22] essentially showed (O, {forests})-colouring to be NP-complete, while Brandstädt et al. [4, Thm. 3] proved the case (O, {P 4 , C 4 } − free graphs).
In contrast, if Π A and Π B can be characterized by a set of connected forbidden subgraphs and Π A is not the set of all edgeless graphs, then (Π A , Π B )-Recognition is NP-hard REF .
Vertex-Partitioning into Fixed Additive Induced-Hereditary Properties Is NP-hard
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Mathematics Computer Science