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Theses

Planar graphs : non-aligned drawings, power domination and enumeration of Eulerian orientations

Abstract : In this thesis, we present results on three different problems concerning planar graphs.We first give some new results on planar non-aligned drawings, i.e. planar grid drawings where vertices are all on different rows and columns.We show that not every planar graph has a non-aligned drawing on an n x n-grid, but we present two algorithms generating a non-aligned polyline drawings on such a grid requiring either n-3 or min((2n-3)/(5),#{separating triangles}+1) bends in total.Concerning non-minimal grids, we give two algorithms drawing a planar non-aligned drawing on grids with area of order n⁴. We also give specific results for 4-connected graphs and nested-triangle graphs.The second topic is power domination in planar graphs. We present a family of graphs with power dominating number γP at least n/6. We then prove that for every maximal planar graph G of order n, γP(G) ≤ (n-2)/4, and we give a constructive algorithm. We also prove that for triangular grids Tk of dimension k with hexagonal-shape border, γP(Tk) = [k/3].Finally, we focus on the enumeration of planar Eulerian orientations. After proposing a new decomposition for these maps, we define subsets and supersets of planar Eulerian orientations with parameter k, generated by looking at the orientations of the last 2k-1 edges around the root vertex.For each set, we give a system of functional equations defining its generating function, and we prove that it is always algebraic.This way, we show that the growth rate of planar Eulerian orientations is between 11.56 and 13.005.
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Claire Pennarun. Planar graphs : non-aligned drawings, power domination and enumeration of Eulerian orientations. Other [cs.OH]. Université de Bordeaux, 2017. English. ⟨NNT : 2017BORD0609⟩. ⟨tel-01591010⟩

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