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Observations on maps and δ-matroids

100%

Richter R.

Discussiones Mathematicae Graph Theory

1996

tom 16

nr 2

197-205

Using a Δ-matroid associated with a map, Anderson et al (J. Combin. Theory (B) 66 (1996) 232-246) showed that one can decide in polynomial time if a medial graph (a 4-regular, 2-face colourable embedded graph) in the sphere, projective plane or torus has two Euler tours that each never cross themselves and never use the same transition at any vertex. With some simple observations, we extend this to the Klein bottle and the sphere with 3 crosscaps and show that the argument does not work in any other surface. We also show there are other Δ-matroids that one can associate with an embedded graph.

Unique factorisation of additive induced-hereditary properties

63%

Farrugia A., Richter R.

Discussiones Mathematicae Graph Theory

2004

tom 24

nr 2

319-343

An additive hereditary graph property is a set of graphs, closed under isomorphism and under taking subgraphs and disjoint unions. Let 𝓟₁,...,𝓟ₙ be additive hereditary graph properties. A graph G has property (𝓟₁∘...∘𝓟ₙ) if there is a partition (V₁,...,Vₙ) of V(G) into n sets such that, for all i, the induced subgraph $G[V_i]$ is in $𝓟_i$. A property 𝓟 is reducible if there are properties 𝓠, 𝓡 such that 𝓟 = 𝓠 ∘ 𝓡 ; otherwise it is irreducible. Mihók, Semanišin and Vasky [8] gave a factorisation for any additive hereditary property 𝓟 into a given number dc(𝓟) of irreducible additive hereditary factors. Mihók [7] 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.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Richter R.

1 Farrugia A.

1 2004

1 1996

Wyniki wyszukiwania

Observations on maps and δ-matroids

Unique factorisation of additive induced-hereditary properties