
Complexity of Domination in Triangulated Plane Graphs
We prove that for a triangulated plane graph it is NPcomplete to determ...
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Vertex partitions of (C_3,C_4,C_6)free planar graphs
A graph is (k_1,k_2)colorable if its vertex set can be partitioned into...
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Complexity of planar signed graph homomorphisms to cycles
We study homomorphism problems of signed graphs. A signed graph is an un...
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On the inversion number of oriented graphs
Let D be an oriented graph. The inversion of a set X of vertices in D co...
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On the degree sequences of dual graphs on surfaces
Given two graphs G and G^* with a onetoone correspondence between thei...
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Computing Cliques and Cavities in Networks
Complex networks have complete subgraphs such as nodes, edges, triangles...
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Königsberg Sightseeing: Eulerian Walks in Temporal Graphs
An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits...
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On the dichromatic number of surfaces
In this paper, we give bounds on the dichromatic number χ⃗(Σ) of a surface Σ, which is the maximum dichromatic number of an oriented graph embeddable on Σ. We determine the asymptotic behaviour of χ⃗(Σ) by showing that there exist constants a_1 and a_2 such that, a_1√(c)/log(c)≤χ⃗(Σ) ≤ a_2 √(c)/log(c) for every surface Σ with Euler characteristic c≤ 2. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane ℕ_1, the Klein bottle ℕ_2, the torus 𝕊_1, and Dyck's surface ℕ_3 are all equal to 3, and that the dichromatic numbers of the 5torus 𝕊_5 and the 10cross surface ℕ_10 are equal to 4. We also consider the complexity of deciding whether a given digraph or oriented graph embedabble in a fixed surface is kdicolourable. In particular, we show that for any surface, deciding whether a digraph embeddable on this surface is 2dicolourable is NPcomplete, and that deciding whether a planar oriented graph is 2dicolourable is NPcomplete unless all planar oriented graphs are 2dicolourable (which was conjectured by NeumannLara).
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