# The Forcing Strong Metric Dimension of a Graph

## Authors

• R Lenin Centre for Research and Post Graduate Studies in Mathematics, Ayya Nadar Janaki Ammal College, Sivakasi, Tamil Nadu, India.
• KM Kathiresan Centre for Research and Post Graduate Studies in Mathematics, Ayya Nadar Janaki Ammal College, Sivakasi Tamil Nadu, India
• Martin Baca Department of Applied Mathematics and Informatics, Technical University, Kosice, Slovakia

## Keywords:

metric, strong metric, forcing strong metric

## Abstract

For any two vertices u, v in a connected graph G, the interval I(u, v) consists of all vertices which are lying in some u − v shortest path in G. A vertex x in a graph G strongly resolves a pair of vertices u, v if either u ∈ I(x, v) or v ∈ I(x, u). A set of vertices W of V (G) is called a strong resolving set if every pair of vertices of G is strongly resolved by some vertex of W. The minimum cardinality of a strong resolving set in G is called the strong metric dimension number of G and it is denoted by sdim(G). For a strong resolving set W of G, a subset S of
W is called the forcing subset of W if W is the unique strong resolving set containing S. The forcing number f(W, sdim(G)) of W in G is the minimum cardinality of a forcing subset for W, while the forcing strong metric dimension, fsdim(G), of G is the smallest forcing number among all strong resolving sets of G. The forcing strong metric dimensions of some well-known graphs are determined. It is shown that for any positive
integers a and b, with 0 ≤ a ≤ b, there is nontrivial connected graph G with sdim(G) = b and fsdim(G) = a if and only if {a, b} not equal to {0, 1}.