scholarly journals A Steiner general position problem in graph theory

2021 ◽  
Vol 40 (6) ◽  
Author(s):  
Sandi Klavžar ◽  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael G. Yero
Keyword(s):  
Graph Theory ◽  
General Position ◽  
Position Problem

A GENERAL POSITION PROBLEM IN GRAPH THEORY

2018 ◽  
Vol 98 (2) ◽  
pp. 177-187 ◽  
Author(s):  
PAUL MANUEL ◽  
SANDI KLAVŽAR
Keyword(s):  
Graph Theory ◽  
Lower Bounds ◽  
General Position ◽  
Upper Bounds ◽  
Np Complete ◽  
Position Problem

The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

scholarly journals The general position problem on Kneser graphs and on some graph operations

2019 ◽  
Author(s):  
Modjtaba Ghorbani ◽  
Sandi Klavžar ◽  
Hamid Reza Maimani ◽  
Mostafa Momeni ◽  
Farhad Rahimi Mahid ◽  
...  
Keyword(s):  
General Position ◽  
Graph Operations ◽  
Kneser Graphs ◽  
Position Problem

The general position problem and strong resolving graphs

2019 ◽  
Vol 17 (1) ◽  
pp. 1126-1135 ◽  
Author(s):  
Sandi Klavžar ◽  
Ismael G. Yero
Keyword(s):  
Connected Graph ◽  
General Position ◽  
Clique Number ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Direct Products ◽  
Strong Product ◽  
Product Graphs ◽  
Complete Bipartite Graphs ◽  
Position Problem

Abstract The general position number gp(G) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp(G) ≥ ω(GSR), where GSR is the strong resolving graph of G, and ω(GSR) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp(G ⊠ H) ≥ gp(G)gp(H), and asked whether the equality holds for arbitrary connected graphs G and H. It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

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