The general position problem and strong resolving graphs

Sandi Klavžar,Ismael G Yero

doi:10.1515/math-2019-0088

Abstract

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.

Highlights

The general position problem was recently and independently introduced in [1, 2]
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
That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and di erent families of strong products, of generalized lexicographic products, and of rooted product graphs

Summary

Introduction

The general position problem was recently and independently introduced in [1, 2]. If G = (V(G), E(G)) is a graph, S ⊆ V(G) is a general position set if no triple of vertices from S lie on a common geodesic in G. The strong resolving graph GSR of G has V(G) as the vertex set, two vertices being adjacent in GSR if they are MMD in G. In the nal section we determine the general position number for di erent rooted product graphs and relate the values with the corresponding clique numbers of strong resolving graphs. A distance-constant partition P is in-transitive if dG(Si , Sk) ≠ dG(Si , Sj) + dG(Sj , Sk) holds for pairwise di erent indices i, j, k ∈ [p] With these concepts, general position sets can be characterized as follows. S ⊆ V(G) is a general position set if and only if the components of S are complete subgraphs, the vertices of which form an in-transitive, distance-constant partition of S.

The lower bound and equality cases

Strong products

Generalized lexicographic products

Rooted product graphs

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