IA Scholar Query: Oscar Defrain
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 28 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440On digraphs without onion star immersions
https://scholar.archive.org/work/hse2assvmzcznpm6km5kecqh2m
The t-onion star is the digraph obtained from a star with 2t leaves by replacing every edge by a triple of arcs, where in t triples we orient two arcs away from the center, and in the remaining t triples we orient two arcs towards the center. Note that the t-onion star contains, as an immersion, every digraph on t vertices where each vertex has outdegree at most 2 and indegree at most 1, or vice versa. We investigate the structure in digraphs that exclude a fixed onion star as an immersion. The main discovery is that in such digraphs, for some duality statements true in the undirected setting we can prove their directed analogues. More specifically, we show the next two statements. There is a function fℕ→ℕ satisfying the following: If a digraph D contains a set X of 2t+1 vertices such that for any x,y∈ X there are f(t) arc-disjoint paths from x to y, then D contains the t-onion star as an immersion. There is a function gℕ×ℕ→ℕ satisfying the following: If x and y is a pair of vertices in a digraph D such that there are at least g(t,k) arc-disjoint paths from x to y and there are at least g(t,k) arc-disjoint paths from y to x, then either D contains the t-onion star as an immersion, or there is a family of 2k pairwise arc-disjoint paths with k paths from x to y and k paths from y to x.Łukasz Bożyk, Oscar Defrain, Karolina Okrasa, Michał Pilipczukwork_hse2assvmzcznpm6km5kecqh2mMon, 28 Nov 2022 00:00:00 GMTConnected greedy colourings of perfect graphs and other classes: the good, the bad and the ugly
https://scholar.archive.org/work/2benuey5vvhmfauyo5y55papr4
The Grundy number of a graph is the maximum number of colours used by the "First-Fit" greedy colouring algorithm over all vertex orderings. Given a vertex ordering σ= v_1,...,v_n, the "First-Fit" greedy colouring algorithm colours the vertices in the order of σ by assigning to each vertex the smallest colour unused in its neighbourhood. By restricting this procedure to vertex orderings that are connected, we obtain connected greedy colourings. For some graphs, all connected greedy colourings use exactly χ(G) colours; they are called good graphs. On the opposite, some graphs do not admit any connected greedy colouring using only χ(G) colours; they are called ugly graphs. We show that no perfect graph is ugly. We also give simple proofs of this fact for subclasses of perfect graphs (block graphs, comparability graphs), and show that no K_4-minor free graph is ugly.Laurent Beaudou and Caroline Brosse and Oscar Defrain and Florent Foucaud and Aurélie Lagoutte and Vincent Limouzy and Lucas Pastorwork_2benuey5vvhmfauyo5y55papr4Thu, 17 Nov 2022 00:00:00 GMTOn objects dual to tree-cut decompositions
https://scholar.archive.org/work/m6jsyzxlejgina2es2nlfbdcfu
Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width.Łukasz Bożyk, Oscar Defrain, Karolina Okrasa, Michał Pilipczukwork_m6jsyzxlejgina2es2nlfbdcfuThu, 11 Aug 2022 00:00:00 GMT