The general position number of a graph $G$ is the size of the largest set $K$ of vertices of $G$ such that no shortest path of $G$ contains three vertices of $K$. In this paper we discuss a related invariant, the monophonic position number, which is obtained from the definition of general position number by replacing `shortest path' with `induced path'. We prove some basic properties of this invariant and determine the monophonic position number of several common types of graphs, including block graphs, unicyclic graphs, complements of bipartite graphs and split graphs. We present an upper bound for the monophonic position numbers of triangle-free graphs and use it to determine the monophonic position numbers of the Petersen graph and the Heawood graph. We then determine realisation results for the general position number, monophonic position number and monophonic hull number and finally find the monophonic position numbers of joins and corona products of graphs.
- Pub Date:
- December 2020
- Mathematics - Combinatorics;
- 05C12 05C69
- The addition of new results mean that this article has grown into two papers, one on structural results and the second on extremal problems. Some corrections made and presentation improved. 19 pages