We determine the minimum cardinality of an identifying code of Kn Kn, the Cartesian product of two cliques of same size. Moreover we show that this code is unique, up to row and column permutations, when n ≥ 5 is odd. If n ≥ 4 is even, we exhibit two distinct optimal identifying codes.

An identifying code in a graph G is a subset of vertices with the property that for each vertex v ∈ V (G), the collection of elements of C at distance at most 1 from v is non-empty and distinct from the collection of any other vertex. We consider the minimum density d(Sk) of an identifying code in the square grid Sk of height k (i.e. with vertex set Z × {1, . . . , k}). Using the Discharging Me...

Let G be a simple, undirected graph with vertex set V . For v ∈ V and r ≥ 1, we denote by BG,r (v) the ball of radius r and centre v. A set C ⊆ V is said to be an r-identifying code in G if the sets BG,r (v) ∩ C , v ∈ V , are all nonempty and distinct. A graph G which admits an r-identifying code is called r-twin-free or r-identifiable, and in this case the smallest size of an r-identifying cod...

An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this paper, we design an algorithm that finds an identifying code in a twin-free graph on n vertices in O(n) binary operations, and returns a failure if the graph is n...

For a directed graph G, a t-identifying code is a subset S ⊆ V (G) with the property that for each vertex v ∈ V (G) the set of vertices of S reachable from v by a directed path of length at most t is both non-empty and unique. A graph is called t-identifiable if there exists a t-identifying code. This paper shows that the de Bruijn graph ~ B(d, n) is 1and 2-identifiable and examines conditions ...

We consider the problem of finding a minimum identifying code in a graph, i.e., a designated set of vertices whose neighborhoods uniquely overlap at any vertex on the graph. This identifying code problem was initially introduced in 1998 and has been since fundamentally connected to a wide range of applications, including fault diagnosis, location detection, environmental monitoring, and connect...

A (1,≤ 2)-identifying code is a subset of the vertex set C of a graph such that each pair of vertices intersects C in a distinct way. This has useful applications in locating errors in multiprocessor networks and threat monitoring. At the time of writing, there is no simply-stated rule that will indicate if a graph is (1,≤ 2)-identifiable. As such, we discuss properties that must be satisfied b...

Code-mixing is a prevalent phenomenon in modern day communication. Though several systems enjoy success in identifying a single language, identifying languages of words in code-mixed texts is a herculean task, more so in a social media context. This paper explores the English-Bengali code-mixing phenomenon and presents algorithms capable of identifying the language of every word to a reasonable...

In this paper the problem of constructing graphs having a (1,≤ `)-identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by Ω ( `2 log ` log n ) . Here we construct graphs on n vertices having a (1,≤ `)-identifying code of cardinality O (`4 log n) for all ` ≥ 2. We derive our construction from a connection between identifying codes and sup...