Identifying codes in line graphs

TL;DR

This paper investigates the size of minimum edge-identifying codes in line graphs, establishing bounds, tightness, and computational complexity, and confirms a conjecture for a subclass of line graphs.

Contribution

It improves known bounds on identifying codes in line graphs, proves the bounds are tight, and shows the NP-completeness of the problem in certain graph classes.

Findings

The lower bound for identifying codes in line graphs is a ext{( ext{n}+1)} ceil, improved to a ext{( ext{n})}

The upper bound a ext{(2|V(G)|-5)} holds with two exceptions

The edge-identifying code problem is NP-complete for planar bipartite graphs with maximum degree 3 and large girth.

Abstract

An identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbours within the code. We study the edge-identifying code problem, i.e. the identifying code problem in line graphs. If $\ID(G)$ denotes the size of a minimum identifying code of an identifiable graph $G$, we show that the usual bound $\ID(G)\ge \lceil\log_2(n+1)\rceil$, where $n$ denotes the order of $G$, can be improved to $\Theta(\sqrt{n})$ in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound $\ID(\mathcal{L}(G))\leq 2|V(G)|-5$, where $\mathcal{L}(G)$ is the line graph of $G$, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud and the first author holds for a subclass of line graphs. Finally, we show that the edge-identifying code problem is NP-complete, even for the class of planar bipartite graphs of maximum degree~3 and arbitrarily large girth.