Solving the kernel perfect problem by (simple) forbidden subdigraphs for digraphs in some families of generalized tournaments and generalized bipartite tournaments

A digraph such that every proper induced subdigraph has a kernel is said to be \emph{kernel perfect} (KP for short) (\emph{critical kernel imperfect} (CKI for short) resp.) if the digraph has a kernel (does not have a kernel resp.). The unique CKI-tournament is $\overrightarrow{C}_3$ and the unique KP-tournaments are the transitive tournaments, however bipartite tournaments are KP. In this paper we characterize the CKI- and KP-digraphs for the following families of digraphs: locally in-/out-semicomplete, asymmetric arc-locally in-/out-semicomplete, asymmetric $3$-quasi-transitive and asymmetric $3$-anti-quasi-transitive $TT_3$-free and we state that the problem of determining whether a digraph of one of these families is CKI is polynomial, giving a solution to a problem closely related to the following conjecture posted by Bang-Jensen in 1998: the kernel problem is polynomially solvable for locally in-semicomplete digraphs.Comment: 13 pages and 5 figure

Vertices with the Second Neighborhood Property in Eulerian Digraphs

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple dicycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but have a vertex of minimum outdegree that has the Second Neighborhood Property.Comment: fixed an error in an earlier version and made structural change

Vertices with the Second Neighborhood Property in Eulerian Digraphs

The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph.Comment: This is the version accepted for publication in Opuscula Mathematic

On the Small Quasi-kernel conjecture

The Chv\'atal-Lov\'asz theorem from 1974 establishes in every finite digraph $G$ the existence of a quasi-kernel, i.e., an independent $2$-out-dominating vertex set. In the same spirit, the Small Quasi-kernel Conjecture, proposed by Erd\H{o}s and Sz\'ekely in 1976, asserts the existence of a quasi-kernel of order at most $|V(G)|/2$ if $G$ does not have sources. Despite repeated efforts, the conjecture remains wide open. This work contains a number of new results towards the conjecture. In our main contribution we resolve the conjecture for all directed graphs without sources containing a kernel in the second out-neighborhood of a quasi-kernel. Furthermore, we provide a novel strongly connected example demonstrating the asymptotic sharpness of the conjecture. Additionally, we resolve the conjecture in a strong form for all directed unicyclic graphs.Comment: 12 pages, 1 figur

Graph Theory

[no abstract available

Clique number of tournaments

We introduce the notion of clique number of a tournament and investigate its relation with the dichromatic number. In particular, it permits defining \dic-bounded classes of tournaments, which is the paper's main topic

Sets as graphs

The aim of this thesis is a mutual transfer of computational and structural results and techniques between sets and graphs. We study combinatorial enumeration of sets, canonical encodings, random generation, digraph immersions. We also investigate the underlying structure of sets in algorithmic terms, or in connection with hereditary graphs classes. Finally, we employ a set-based proof-checker to verify two classical results on claw-free graph