On the minimum doubly resolving set problem in line graphs

TL;DR

This paper investigates the minimum size of a doubly resolving set in line graphs, proving NP-hardness, establishing bounds related to maximum degree, and exactly determining the value for trees.

Contribution

It introduces the problem of finding the minimum doubly resolving set in line graphs, proves its NP-hardness, derives bounds, and solves it exactly for trees.

Findings

Computing the minimum DRS in line graphs is NP-hard.

Bounds for the minimum DRS depend on maximum degree and number of vertices.

Exact value of the minimum DRS is determined for trees.

Abstract

Given a connected graph $G$ with at least three vertices, let $d_G(u,v)$ denote the distance between vertices $u,v\in V(G)$. A subset $S\subseteq V$ is called a doubly resolving set (DRS) of $G$ if for any two distinct vertices $u, v \in V(G)$, there exists a pair $\{x,y\}\subseteq S$ such that $d_G(u,x)-d_G(u,y)\neq d_G(v,x)-d_G(v,y)$. This paper studies the minimum cardinality of a DRS in the line graph of $G$, denoted by $\Psi(L(G))$. First, we prove that computing $\Psi(L(G))$ is NP-hard, even when $G$ is a bipartite graph. Second, we establish that $\lceil \log_2 (1+\Delta(G))\rceil \le \Psi(L(G)) \le |V(G)| - 1$ holds for all $G$ with maximum degree $\Delta(G)$, and show that both inequalities are tight. Finally, we determine the exact value of $\Psi(L(G))$ provided $G$ is a tree.