Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments. A feedback arc set (fas) in a digraph D = (V,A) is a set F of arcs such that D \F is acyclic. The size of a minimum feedback arc set of D is denoted by mfas(D). A classical result of Lawler and Karp [5] asserts that finding a minimum feedback arc set in a digraph...

The notion of a competition graph was introduced by J. E. Cohen in 1968. The competition graph C(D) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that (x, v) and (y, v) are arcs of D. For any graph G, G together with sufficiently many new isolated vertices is...

For two positive integers k and `, a (k × `)-spindle is the union of k pairwise internally vertexdisjoint directed paths with ` arcs between two vertices u and v. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain th...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

We consider the one-variable characteristic polynomial p(G; ) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe/cients and the degree of p(G; ). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G; ) gives th...

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f (d, n) (fa(d, n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d, n). This paper proves that for any integers d 2 and n 1 f (d, n)= ⎪⎪⎪⎨ ⎪⎪⎪⎩ d for n= 1, ( )(n) n + ( )(n− 1) n− 1 for 2 n 7, dn n +...

Given a digraph D, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We in...

Given an edge-weighted tree T with leaf set X, define the weight of a subset S of X as the sum of the edge-weights of the minimal subtree of T connecting the elements in S. It is known that the problem of selecting subsets of X of a given size to maximize this weight can be solved using a greedy algorithm. This optimization problem arises in conservation biology where the weight is referred to ...

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there is a vertex x in D such that (u, x) and (v, x) are arcs of D. In this paper, we show that the competition graphs of doubly partial orders are interval graphs. We also show that an interval graph together with enough isolated vertices is the competit...

We consider monotonicity problems for graph searching games. Variants of these games – defined by the type of moves allowed for the players – have been found to be closely connected to graph decompositions and associated width measures such as pathor tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowin...