Can the vertices of a graph 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? The NP-completeness of these problems are just 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 Pand Q-recognition are both in NP. This proves a conjecture of Kratochvíl and Schiermeyer.
An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let P 1 , . . . , P n be additive hereditary graph properties. A graph G has property (P 1 • · · · • P n ) if there is a partition (V 1 , . . . , V n ) of V (G) into n sets such that, for all i, the induced subgraphSemanišin and Vasky [J. Graph Theory 33 (2000), 44-53] gave a factorisation for any additive hereditary property P into a given number dc(P) of irreducible additive hereditary factors. Mihók [Discuss. Math. Graph Theory 20 (2000), 143-153] gave a similar factorisation for properties that are additive and induced-hereditary (closed under taking induced-subgraphs and disjoint unions).Their results left open the possiblity of different factorisations, maybe even with a different number of factors; we prove here that the given factorisations are, in fact, unique.
We prove three results conjectured or stated by Chartrand and Zhang [European J. ]: a connected graph has orientations with different geodetic numbers, orientations with different hull numbers, and, if there are no end-vertices, orientations with different convexity numbers. The proof of the first result is a correction of Chartrand and Zhang's proof, and allows for an easy proof of the second result. The third result says roughly that graphs without end-vertices can be oriented anti-transitively.
A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If P and Q are properties, the product P • Q consists of all graphs G for which there is a partition of the vertex set of G into (possibly empty) subsets A and B with G[A] ∈ P and G[B] ∈ Q. A property is * The results presented here are part of the first author's Ph.D. thesis, written under the supervision of the second author. Jim Geelen suggested one of the main results of this paper.† The first author's doctoral studies in Canada were fully funded by the Canadian government through a Canadian Commonwealth Scholarship.‡ Research supported by NSERC.§ Research supported in part by Slovak VEGA Grant 1/0424/03. reducible if it is the product of two other properties, and irreducible otherwise.We completely describe the few reducible induced-hereditary properties that have a unique factorisation into irreducibles. Analogs of compositive and additive induced-hereditary properties are introduced and characterised in the style of Scheinerman [Discrete Math. 55 (1985) 185-193]. One of these provides an alternative proof that an additive hereditary property factors into irreducible additive hereditary properties.
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