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A GENERAL POSITION PROBLEM IN GRAPH THEORY

Part of: Graph theory Theory of computing

Published online by Cambridge University Press:  18 July 2018

PAUL MANUEL Open the ORCID record for PAUL MANUEL [Opens in a new window]  and
SANDI KLAVŽAR
PAUL MANUEL*
Affiliation:
Department of Information Science, College of Computing Science and Engineering, Kuwait University, Kuwait email [email protected]
SANDI KLAVŽAR
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Slovenia Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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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.

Keywords

general position problemisometric subgraphpackingindependence numbercomputational complexity

MSC classification

Primary: 05C12: Distance in graphs
Secondary: 05C70: Factorization, matching, partitioning, covering and packing 68Q25: Analysis of algorithms and problem complexity
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 177 - 187
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported and funded by Kuwait University, Research Project No. QI 02/17.

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