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In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic(More)
We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erd˝ os, we prove that, if the minimum degree of G is at least 17|V (G)|/20, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete(More)
We prove that every graph G has a vertex partition into a cycle and an anticycle (a cycle in the complement of G). Emptyset, singletons and edges are considered as cycles. This problem was posed by Lehel and shown to be true for very large graphs by Luczak, Rödl and Szemerédi [7], and more recently for large graphs by Allen [1]. Many questions deal with the(More)
We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G with at most 5k 2 +k vertices and an integer k such that G has a feedback vertex set of size at most k iff G has a feedback vertex set of size at most k. This result improves a previous O(k 11) kernel of Burrage et al. [6], and a more(More)
We prove that with three exceptions, every tournament of order n contains each oriented path of order n. The exceptions are the antidirected paths in the 3-cycle, in the regular tournament on 5 vertices, and in the Paley tournament on 7 vertices. Tournaments are very rich structures and many questions deal with their subgraphs. In particular, much work has(More)
We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its ®rst outneighborhood. Moreover, we exhibit two such vertices provided that the tournament has no dominated vertex. The proof makes use of median orders. A second application of median orders is that every tournament(More)
In 1963, Tibor Gallai [9] asked whether every strongly connected directed graph D is spanned by α directed circuits, where α is the stability of D. We give a proof of this conjecture. In this paper, circuits of length two are allowed. Since loops and multiple arcs play no role in this topic, we will simply assume that our digraphs are loopless and simple. A(More)
Adapting the method introduced in Graph Minors X, we propose a new proof of the duality between the bramble-number of a graph and its tree-width. Our approach is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The proof does not rely on Menger's theorem, and thus greatly generalises(More)