On the minimum doubly resolving set problem in line graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918313043427328 |
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| author | Ye, Qingjie |
| author_facet | Ye, Qingjie |
| contents | 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 $Ψ(L(G))$. First, we prove that computing $Ψ(L(G))$ is NP-hard, even when $G$ is a bipartite graph. Second, we establish that $\lceil \log_2 (1+Δ(G))\rceil \le Ψ(L(G)) \le |V(G)| - 1$ holds for all $G$ with maximum degree $Δ(G)$, and show that both inequalities are tight. Finally, we determine the exact value of $Ψ(L(G))$ provided $G$ is a tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21580 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the minimum doubly resolving set problem in line graphs Ye, Qingjie Combinatorics Optimization and Control 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 $Ψ(L(G))$. First, we prove that computing $Ψ(L(G))$ is NP-hard, even when $G$ is a bipartite graph. Second, we establish that $\lceil \log_2 (1+Δ(G))\rceil \le Ψ(L(G)) \le |V(G)| - 1$ holds for all $G$ with maximum degree $Δ(G)$, and show that both inequalities are tight. Finally, we determine the exact value of $Ψ(L(G))$ provided $G$ is a tree. |
| title | On the minimum doubly resolving set problem in line graphs |
| topic | Combinatorics Optimization and Control |
| url | https://arxiv.org/abs/2601.21580 |