Research project is (co) funded by the Slovenian Research Agency.
UL Member: Faculty of Mathematics and Physics
Code: N1-0285
Project: Metric Problems in Graphs and Hypergraphs
Period: 1. 1. 2023 - 31. 12. 2025
Range per year: 0,6 FTE, category: C
Head: Sandi Klavžar
Research activity: Natural sciences and mathematics
Research Organisations, Researchers and Citations for bibliographic records
Project description:
The aim of the project is to make important contributions to the metric graph theory and to extend the theory to hypergraphs. We will first solve the set open problems on graphs and then investigate whether they can be extended in one way or another to hypergraphs or to special hypergraphs that allow such extensions as, for example, linear hypergraphs. Specific goals of the project can be combined into four rounded bigger objectives.
(1) Investigation of the Wiener index. As the Wiener index can be expressed as one half of the transmissions of its vertices, we will pay a special attention to the transmission of vertices. In particular, graphs in which vertices have pairwise different transmissions will be investigated. Distance based invariants which are Wiener-like will also be closely studied. Since the cut method turned out to be extremely useful for the investigation of the Wiener index of graphs as well as additional distance-based graph invariants, our goal is to extend this theory to hypergraphs.
(2) Investigation of the metric dimension. We will investigate different variants of the classical metric dimension including the edge metric dimension and the mixed metric dimension. Special attention will be given to the multiset metric dimension. On hypergraphs we will investigate both the classical metric dimension as well as variants of it and try to extend the obtained results for the edge metric dimension and the mixed metric dimension from graph to hypergraphs.
(3) Investigation of the strong (edge) geodetic number. We plan to extend known results for the strong geodetic number and the strong edge geodetic number for Cartesian product graphs. We plan to investigate these two invariants on other standard graph products and other families of graphs. The strong edge geodetic problem will be investigated on complete bipartite graphs. For general graphs we aim to find sharp upper and lower bounds on the (edge) strong geodetic number. The strong geodetic number and the edge strong geodetic number will be compared with each other. We will extend the newly obtained results as well as known results from graphs to hypergraphs.
(4) Investigation of the general position problem. We will continue research of the general position problem on vertices and initiate the research of the edge version of the problem. We will extend both problems to hypergraphs by extending known results whenever possible as well as extending the new results we will obtain for graphs. In particular, we plan to characterize general position sets in hypergraphs and to investigate the general position number and the edge general position number on hypergraphs with a tree structure and on hypergraphs with a product structure.
