On the number of nonseparating vertices in strongly connected local tournaments

On the number of nonseparating vertices in strongly connected in-tournaments

A digraph without loops, multiple arcs and directed cycles of length two is called an in-tournament if the set of in-neighbors of every vertex induces a tournament. A local tournament is an in-tournament such that the set of out-neighbors of every vertex induces a tournament as well. Let p ≥ 2 be an integer and let T be a strongly connected tournament such that every vertex has at least p posit...

Nonseparating vertices in tournaments with large minimum degree

Maximum Distance Between Slater Orders and Copeland Orders of Tournaments

Given a tournament T = (X, A), we consider two tournament solutions applied to T: Slater’s solution and Copeland’s solution. Slater’s solution consists in determining the linear orders obtained by reversing a minimum number of directed edges of T in order to make T transitive. Copeland’s solution applied to T ranks the vertices of T according to their decreasing out-degrees. The aim of this pap...

The number of C3-free vertices on 3-partite tournaments

Let T be a 3-partite tournament. We say that a vertex v is −→ C3 -free if v does not lie on any directed triangle of T . Let F3(T ) be the set of the −→ C3 -free vertices in a 3-partite tournament and f3(T ) its cardinality. In this paper we prove that if T is a regular 3-partite tournament, then F3(T )must be contained in one of the partite sets of T . It is also shown that for every regular 3...

Local Out-Tournaments with Upset Tournament Strong Components I: Full and Equal {0, 1}-Matrix Ranks

A digraph D is a local out-tournament if the outset of every vertex: is a tournament. Here, we use local out-tournaments, whose strong components are upset tournaments, to explore the corresponding ranks of the adjacency matrices. Of specific interest is the out-tournament whose adjacency matrix has boolean, nonnegative integer, term, and real rank all equal to the number of vertices, n. Corres...