Gabriel Semanišin

• Prof., Dr., PhD.
• Head of Faculty at University of Pavol Jozef Šafárik

The inspiration for writing about the Košice mathematics school came from conversations with the professor. Lev Bukovský and prof. Stanislav Jendroľ during the preparation and creation of publications about them, in the Personalities of Slovak Mathematics Edition framework. In this publication we present other important events, the Conference of Ko...

Hate speech detection can be seen as the process of categorizing and identifying speech or written content that promotes discrimination, or violence towards individuals or groups. It can include various computational algorithms, natural language processing methods, and machine learning algorithms to automatically analyze and classify text data for...

Reactive power is an important part of the electric power systems in order to rotate machines or to transmit active power by transmission lines. However, an excess of reactive power in the electrical systems can in- crease the risk of failure of the transmission system. We present an automatic reactive power classification on multifamily residentia...

A set $S$ of vertices of a graph $G$ is a geodesic transversal of $G$ if every maximal geodesic of $G$ contains at least one vertex of $S$. The minimum cardinality of a geodesic transversal of $G$ is denoted by $\txgt(G)$ and is called geodesic transversal number. For two graphs $G$ and $H$ we deal with the behavior of this invariant for the lexico...

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S⊆V(G) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ(G). We sho...

Vertex cover number, which is one of the most basic graph invariants, can be viewed as the smallest number of vertices that hit (or that belong to) every subgraph K 2 in a graph G. In this paper, we consider the next two smallest cases of connected graphs, which are the path P 3 and the cycle C 3 ; the problem is to minimize the set of vertices tha...

The total generalised colourings considered in this paper are colourings of the vertices and of the edges of graphs satisfying the following conditions: • each set of vertices of the graph which receive the same colour induces an m-degenerate graph,• each set of edges of the graph which receive the same colour induces an n-degenerate graph, and• in...

We consider ( ψk-γk-1)-perfect graphs, i.e., graphs G for which ψk(H) = γk-1(H) for any induced subgraph H of G, where ψk and γk-1 are the k-path vertex cover number and the distance (k - 1)-domination number, respectively. We study ( ψk-γk-1)-perfect paths, cycles and complete graphs for k ≥ 2. Moreover, we provide a complete characterisation of (...

A semi-matching in a bipartite graph \(G = (U, V, E)\) is a set of edges \(M \subseteq E\) such that each vertex in U is incident to exactly one edge in M, i.e., \(deg_M(u)=1\) for each \(u \in U\). An optimal semi-matching is a semi-matching with the minimal value of the cost function \(\sum _{v \in V} \frac{deg_M(v) \cdot (deg_M(v)+1)}{2}\). Expl...

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. The cardinality of a minimum k-path vertex cover is called the k-path vertex cover number of a graph G, denoted by ψk(G). It is known that the minimum k-path vertex cover problem (k-PVCP) is NP-hard for all k ≥ 2. I...

A subset $S$ of vertices of a graph $G$ is called a vertex $k$-path cover if every path of order $k$ in $G$ contains at least one vertex from $S$. Denote by $ψ_k(G)$ the minimum cardinality of a vertex $k$-path cover in $G$. In this paper, an upper bound for $ψ_3$ in graphs with a given average degree is presented. A lower bound for $ψ_k$ of regula...

An (f,g)-semi-matching in a bipartite graph G= (U∪V,E) is a set of edges M⊆E such that each vertex u∈U is incident with at most f(u) edges of M, and each vertex v∈V is incident with at most g(v) edges of M. In this paper we give an algorithm that for a graph with n vertices and m edges, n≤m, constructs a maximum (f,g)-semi-matching in running time...

The problem of finding an optimal semi-matching is a generalization of the problem of finding classical matching in bipartite graphs. A semi-matching in a bipartite graph G = (U, V, E) with n vertices and m edges is a set of edges M ⊆ E, such that each vertex in U is incident to at most one edge in M. An optimal semi-matching is a semi-matching wit...

An $(f,g)$-semi-matching in a bipartite graph $G=(U \cup V,E)$ is a set of edges $M \subseteq E$ such that each vertex $u\in U$ is incident with at most $f(u)$ edges of $M$, and each vertex $v\in V$ is incident with at most $g(v)$ edges of $M$. In this paper we give an algorithm that for a graph with $n$ vertices and $m$ edges, $n\le m$, constructs...

The concept of an object-system is a common generalization of simple graph, digraph and hypergraph. In the theory of generalized colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known fundamental theorem of aithmetics. The purpose of this paper is to pr...

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by \psi_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining \psi_k(G) is NP-hard for each k \geq 2, while for trees the problem can be solved in linear time...

An edge-ordering of a graph G=(V,E) is a one-to-one function f from E to a subset of the set of positive integers. A path P in G is called an f-ascent if f increases along the edge sequence of P. The heighth(f) of f is the maximum length of an f-ascent in G.In this paper we deal with computational problems concerning finding ascents in graphs. We p...

Let τ(G) denote the number of vertices in a longest path in a graph G=(V,E). A subset K of V is called a Pn-kernel of G if τ(G[K])≤n−1 and every vertex v∈V∖K is adjacent to an end-vertex of a path of order n−1 in G[K]. It is known that every graph has a Pn-kernel for every positive integer n≤9. R. Aldred and C. Thomassen in [R.E.L. Aldred, C. Thoma...

abs A graph property is any isomorphism closed class of simple graphs. For a simple finite graph H, let → H denote the class of all simple countable graphs that admit homomorphisms to H, such classes of graphs are called hom-properties. Given a graph property P, a graph G ε P is universal in P if each member of P is isomorphic to an induced subgrap...

One of the classical results in graph theory states that every two longest path in a connected graph (also called detours) have a vertex in common. The corresponding problem for three longest paths in a graph is still unsolved. For the oriented graphs one can easily construct a graph having k non-intersecting detours for any integer k≥2. But the si...

We consider a problem of computing the maximal value associated to the nodes of a network in the model of unknown symmetric radio network with availability of collision detection. We assume that the nodes have no initial knowledge about the network topology, number of nodes and even they have no identifiers. The network contains one distinguished n...

The detour order (of a vertex v) of a graph G is the order of a longest path (beginning at v). The detour sequence of G is a sequence consisting of the detour orders of its vertices. A graph is called a detour graph if its detour sequence is constant. The detour deficiency of a graph G is the difference between the order of G and its detour order....

The product P Q of graph properties P;Q is a class of all graphs having a vertex-partition into two parts inducing subgraphs with properties P and Q, respectively. For a graph invariant ' and a graph property P we define '(P) as the minimum of '(F ) taken over all minimal forbidden subgraphs F of P. An invariant of graph properties ' is said to be...

If T=(V,E) is a tree then –T denotes the additive hereditary property consisting of all graphs that does not contain T as a subgraph. For an arbitrary vertex v of T we deal with a partition of T into two trees T1, T2, so that V(T1)∩V(T2)={v}, V(T1)∪(T2)=V(T), E(T1)∩E(T2)=∅, E(T1)∪E(T2)=E(T), T[V(T1)\{v}]⊆E(T1) and T[V(T2)\{v}]⊆E(T2). We call such a...

In the theory of generalised colourings of graphs, the Unique Factorization Theorem (UFT) for additive induced-hereditary properties of graphs provides an analogy of the well-known Fundamental Theorem of Arithmetics. The purpose of this paper is to present a new, less complicated, proof of this theorem that is based on Formal Concept Analysis. The...

A k-ranking of a graph G = (V, E) is a mapping ϕ: V → {1, 2, ..., k} such that each path with end vertices of the same colour c contains an internal vertex with colour greater than c. The ranking number of a graph G is the smallest positive integer k admitting a k-ranking of G. In the on-line version of the problem, the vertices v 1, v 2, ..., v n...

We consider deterministic radio broadcasting in radio networks whose nodes have full topological information about network and the reachability graph of a network is κ-degenerate. The goal is to design a polynomial algorithm which produces a fast radio broadcast schedule with respect to a reachability graph ∈ and a source s e V (G). The length of p...

The original article to which this Erratum refers was published in Journal of Graph Theory 49:11–27 . No Abstract.

An Erratum has been published for this article in Journal of Graph Theory 50:261, 2005. A graph property (i.e., a set of graphs) is hereditary (respectively, induced-hereditary) if it is closed under taking subgraphs (resp., induced-subgraphs), while the property is additive if it is closed under disjoint unions. If and are properties, the product...

Let (Ma,⊆) and (La,⊆) be the lattices of additive induced-hereditary properties of graphs and additive hereditary properties of graphs, respectively. A property R∈Ma (∈La) is called a minimal reducible bound for a property P∈Ma (∈La) if in the interval (P,R) of the lattice Ma (La) there are only irreducible properties. The set of all minimal reduci...

A graph property (i.e., a set of graphs) is induced-hereditary or additive if it is closed under taking induced-subgraphs or disjoint unions. If $\cP$ and $\cQ$ are properties, the product $\cP \circ \cQ$ consists of all graphs $G$ for which there is a partition of the vertex set of $G$ into (possibly empty) subsets $A$ and $B$ with $G[A] \in \cP$...

We show that additive induced-hereditary properties of coloured hypergraphs can be uniquely factorised into irreducible factors. Our constructions and proofs are so general that they can be used for arbitrary concrete categories of combinatorial objects; we provide some examples of such combinatorial objects.

A property P of infinite graphs is said to be of finite character if a graph G has property P if and only if every finite vertex-induced subgraph of G has property P. Using a generalization of the well-known Erd˝os-de Bruijn Theo- rem for arbitrary properties of finite character, we present a proof of the Unique Factorization Theorem for additive p...

A hereditary property of graphs is any class of graphs closed under isomorphism and subgraphs. Let 1, 2,…, n be hereditary properties of graphs. We say that a graph G has property 1°2°···°n if the vertex set of G can be partitioned into n sets V1, V2,…, Vn such that the subgraph of G induced by Vi belongs to i; i = 1, 2,…, n. A hereditary property...

We translate Ramsey-type problems into the language of decomposable hereditary properties of graphs. We prove a distributive law for reducible and decomposable properties of graphs. Using it we establish some values of graph theoretical invariants of decomposable properties and show their correspondence to generalized Ramsey numbers.

For a simple graph H, →H denotes the class of all graphs that admit homomorphisms to H (such classes of graphs are called hom-properties). We investigate hom-properties from the point of view of the lattice of hereditary properties. In particular, we are interested in characterization of maximal graphs belonging to →H. We also provide a description...

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure , is emphasized. Many related topics, including questions on the comple...

In this paper we survey results and open problems on the structure of additive and hereditary properties of graphs. The important role of vertex partition problems, in particular the existence of uniquely partitionable graphs and reducible properties of graphs in this structure , is emphasized. Many related topics, including questions on the comple...

A graph G is called k-degenerate if every subgraph of G has a vertex of degree at most k. A k-degenerate graph G is maximal k-degenerate if for every edge e ϵ E(G), G + e is not k-degenerate. Necessary and sufficient conditions for the sequence II = (d1, d2, ⃛, dp) to be a degree sequence of a maximal k-degenerate graph G are presented. © 1995 John...

We consider deterministic radio broadcasting in radio networks whose nodes have full topological infor- mation about network and the reachability graph of a net- work is k-degenerate. The goal is to design a polynomial al- gorithm which produces a fast radio broadcast schedule with respect to a reachability graph G and a source s 2 V (G). The lengt...