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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)
A tournament T = (V, A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is(More)
Given a graph G = (V, E) and a positive integer k, the Proper Interval Completion problem asks whether there exists a set F of at most k pairs of (V × V) \ E such that the graph H = (V, E ∪ F) is a proper interval graph. The Proper Interval Completion problem finds applications in molecular biology and genomic research [16, 24]. First announced by Kaplan,(More)
In this paper we consider the Maximum Independent Set problem (MIS) on B1-EPG graphs. EPG (for Edge intersection graphs of Paths on a Grid) was introduced in [9] as the class of graphs whose vertices can be represented as simple paths on a rectangular grid so that two vertices are adjacent if and only if the corresponding paths share at least one edge of(More)
A graph G = (V, E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u, v) ∈ E iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification.(More)
The Bermond-Thomassen conjecture states that, for any positive integer r, a digraph of minimum out-degree at least 2r − 1 contains at least r vertex-disjoint directed cycles. Thomassen proved that it is true when r = 2, and very recently the conjecture was proved for the case where r = 3. It is still open for larger values of r, even when restricted to(More)
A k-digraph is a digraph in which every vertex has outdegree at most k. A (k ∨ l)-digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ(k) (resp. Φ ∨ (k, l)) of the arc-chromatic number over the k-digraphs (resp. (k ∨ l)-digraphs). El-Sahili [3] showed that Φ ∨(More)