On the minimum doubly resolving set problem in line graphs

TL;DR

This work analyzes $\Psi(L(G))$, the minimum size of a doubly resolving set in the line graph $L(G)$. It proves NP-hardness of computing $\Psi(L(G))$ even for line graphs of bipartite graphs via a 3-dimensional matching reduction, and it establishes tight bounds $\lceil \log_2 (1+\Delta(G))\rceil \le \Psi(L(G)) \le |V(G)|-1$, with these bounds shown to be tight. For trees, the paper determines the exact value $\Psi(L(T))=\sigma(T)-ex'(T)$ and provides a linear-time algorithm to compute a minimum DRS of $L(T)$. Together, these results advance understanding of doubly resolving sets in line graphs, offering both complexity insights and precise structural characterizations with potential implications for diffusion source localization and graph discrimination in networks.

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 $Ψ(L(G))$. 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.

Figures (1)

• Figure 1: The graph $A_5$

Theorems & Definitions (10)

• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• lemma thmcounterlemma
• theorem thmcountertheorem
• theorem thmcountertheorem
• lemma thmcounterlemma
• proof : Proof of Theorem \ref{['main']} (Lower bound)
• proposition thmcounterproposition