arXiv subjects

Jan Bok

Publications and source records attributed to Jan Bok.

25 records · Page 2Linked to original sources

arXiv2019

On Extremal Graphs of Weighted Szeged Index

An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($\textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular, we proved that the star is a tree having the maximal $\textrm{sz}(G)$. Finding a tree with the minimal $\textrm{sz}(G)$ is not an easy task to be done. Here, we present the minimal trees up to 25 vertices obtained by computer and describe the regularities which retain in them. Our preliminary computer tests suggest that a tree with the minimal $\textrm{sz}(G)$ is also the connected graph of the given order that attains the minimal weighted Szeged index. Additionally, it is proven that among the bipartite connected graphs the complete balanced bipartite graph $K_{\left\lfloor n/2\right\rfloor\left\lceil n/2 \right\rceil}$ attains the maximal $\textrm{sz}(G)$\,. We believe that the $K_{\left\lfloor n/2\right\rfloor\left\lceil n/2 \right\rceil}$ is a connected graph of given order that attains the maximum $\textrm{sz}(G)$.

math.CO↗

arXiv2018

A note on simultaneous representation problem for interval and circular-arc graphs

In this short note, we show two NP-completeness results regarding the \emph{simultaneous representation problem}, introduced by Lubiw and Jampani. The simultaneous representation problem for a given class of intersection graphs asks if some $k$ graphs can be represented so that every vertex is represented by the same interval in each representation. We prove that it is NP-complete to decide this for the class of interval and circular-arc graphs in the case when $k$ is a part of the input and graphs are not in a sunflower position.

cs.DM↗

arXiv2018

On convexity and solution concepts in cooperative interval games

Cooperative interval game is a cooperative game in which every coalition gets assigned some closed real interval. This models uncertainty about how much the members of a coalition get for cooperating together. In this paper we study convexity, core and the Shapley value of games with interval uncertainty. Our motivation to do so is twofold. First, we want to capture which properties are preserved when we generalize concepts from classical cooperative game theory to interval games. Second, since these generalizations can be done in different ways, mainly with regard to the resulting level of uncertainty, we try to compare them and show their relation to each other.

cs.GT↗

arXiv2018

Characterizing subclasses of cover-incomparability graphs by forbidden subposets

In this paper we continue investigations of cover-incomparability graphs of finite partially ordered sets (see \cite{Bres,Bres2,Bres3,Bres4} and \cite{Max,MaxDH}). We consider in some detail the distinction between cover-preserving subsets and isometric subsets of a partially ordered set. This is critical to understanding why forbidden subposet characterizations of certain classes of cover-incomparability graphs in \cite{Bres} and \cite{Bres3} are not valid as presented. Here we provide examples, investigate the root of the difficulties, and formulate and prove valid revisions of these characterizations.

math.CO↗

arXiv2018

Selection-based Approach to Cooperative Interval Games

Cooperative interval games are a generalized model of cooperative games in which the worth of every coalition corresponds to a closed interval representing the possible outcomes of its cooperation. Selections are all possible outcomes of the interval game with no additional uncertainty. We introduce new selection-based classes of interval games and prove their characterization theorems and relations to existing classes based on the interval weakly better operator. We show new results regarding the core and imputations and examine a problem of equivalence for two different versions of the core, the main stability solution of cooperative games. Finally, we introduce the definition of strong imputation and strong core as universal solution concepts of interval games.

math.OC↗

arXiv2018

Algorithmic aspects of $M$-Lipschitz mappings of graphs

$M$-Lipschitz mappings of graphs (or equivalently graph-indexed random walks) are a generalization of standard random walk on $\mathbb{Z}$. For $M \in \N$, an \emph{$M$-Lipschitz mapping} of a connected rooted graph $G = (V,E)$ is a mapping $f: V \to \Z$ such that root is mapped to zero and for every edge $(u,v) \in E$ we have $|f(u) - f(v)| \le M$. We study two natural problems regarding graph-indexed random walks. - Computing the maximum range of a graph-indexed random walk for a given graph. - Deciding if we can extend a partial GI random walk into a full GI random walk for a given graph. We show that both these problems are polynomial-time solvable and we show efficient algorithms for them. To our best knowledge, this is the first algorithmic treatment of Lipschitz mappings of graphs. Furthermore, our problem of extending partial mappings is connected to the problem of \emph{list homomorphism} and yields a better run-time complexity for a specific family of its instances.

math.CO↗

arXiv2018

Graph-indexed random walks on special classes of graphs

We investigate the paramater of the average range of $M$-Lipschitz mapping of a given graph. We focus on well-known classes such as paths, complete graphs, complete bipartite graphs and cycles and show closed formulas for computing this parameter and also we conclude asymptotics of this parameter on these aforementioned classes.

math.CO↗