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

More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

Independent sets and non-augmentable paths in generalizations of tournaments

AbstractWe study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arc-local tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly internally and finally non-augmentable paths as well as a result that relates the degree of vertices and the length of longest paths. A short survey is included in the introduction

A tree representation for P4-sparse graphs

AbstractA graph G is P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize both the class of cographs and the class of P4-reducible graphs. We give several characterizations for P4-sparse graphs and show that they can be constructed from single-vertex graphs by a finite sequence of operations. Our characterization implies that the P4-sparse graphs admit a tree representation unique up to isomorphism. Furthermore, this tree representation can be obtained in polynomial time

Parameterized Enumeration of Neighbour Strings and Kemeny Aggregations

In this thesis, we consider approaches to enumeration problems in the parameterized complexity setting. We obtain competitive parameterized algorithms to enumerate all, as well as several of, the solutions for two related problems Neighbour String and Kemeny Rank Aggregation. In both problems, the goal is to find a solution that is as close as possible to a set of inputs (strings and total orders, respectively) according to some distance measure. We also introduce a notion of enumerative kernels for which there is a bijection between solutions to the original instance and solutions to the kernel, and provide such a kernel for Kemeny Rank Aggregation, improving a previous kernel for the problem. We demonstrate how several of the algorithms and notions discussed in this thesis are extensible to a group of parameterized problems, improving published results for some other problems

Set- and Graph-theoretic Investigations in Abstract Argumentation

Abstract argumentation roots to similar parts in philosophy, linguistics and artificial intelligence. The core (syntactic) notions of argument and attack are commonly visualized via digraphs, as nodes and directed edges, respectively. Semantic evaluation functions then provide a meaning of acceptance (i.e. acceptable sets of arguments also called extensions) for any such abstract argumentation structure. In this thesis, for the very first time, we tackle the questions of acceptance and conflict from a graph- and set-theoretic point of view. We elaborate on the interspace between syntactic conflict/independence (defined by attack structure) and their semantic counterparts (defined by joint acceptance of arguments). Graph theory regards the filters and techniques we use to, respectively, categorize and describe abstract argumentation structures. Set theory regards the issues we have to deal with particularly for non-finite argument sets. For argumentation in the arbitrarily infinite case this thesis can and should be seen as reference work. For the matter of conflicts in abstract argumentation we further provide a solid base and formal framework for future research. All in all, this is a mathematicians view on abstract argumentation, deepening the field of conception and widening the angle of applicability

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum