Identifying codes from the spectrum of a graph or digraph

Peer ReviewedPostprint (author's final draft

On identifying codes in partial linear spaces

Let (P,L, I) be a partial linear space and X ⊆ P ∪ L. Let us denote by (X)I = x∈X{y : yIx} and by [X] = (X)I ∪ X. With this terminology a partial linear space (P,L, I) is said to admit a (1,≤ k)-identifying code if the sets [X] are mutually different for all X ⊆ P ∪L with |X| ≤ k. In this paper we give a characterization of k-regular partial linear spaces admitting a (1,≤ k)-identifying code. Equivalently, we give a characterization of k-regular bipartite graphs of girth at least six admitting a (1,≤ k)-identifying code. That is, k-regular bipartite graphs of girth at least six admitting a set C of vertices such that the sets N[x] ∩ C are nonempty and pairwise distinct for all vertex x ∈ X where X is a subset of vertices of |X| ≤ k. Moreover, we present a family of k-regular partial linear spaces on 2(k−1)2+k points and 2(k − 1)2 + k lines whose incidence graphs do not admit a (1,≤ k)-identifying code. Finally, we show that the smallest (k; 6)-graphs known up to now for k − 1 not a prime power admit a (1,≤ k)-identifying code.Peer Reviewe

Sufficient conditions for a digraph to admit a (1,=l)-identifying code

A (1, = `)-identifying code in a digraph D is a subset C of vertices of D such that all distinct subsets of vertices of cardinality at most ` have distinct closed in-neighbourhoods within C. In this paper, we give some sufficient conditions for a digraph of minimum in-degree d - = 1 to admit a (1, = `)- identifying code for ` ¿ {d -, d- + 1}. As a corollary, we obtain the result by Laihonen that states that a graph of minimum degree d = 2 and girth at least 7 admits a (1, = d)-identifying code. Moreover, we prove that every 1-in-regular digraph has a (1, = 2)-identifying code if and only if the girth of the digraph is at least 5. We also characterize all the 2-in-regular digraphs admitting a (1, = `)-identifying code for ` ¿ {2, 3}.Peer ReviewedPostprint (author's final draft

Characterizing identifying codes from the spectrum of a graph or digraph

A -identifying code in digraph D is a dominating subset C of vertices of D, such that all distinct subsets of vertices of D with cardinality at most l have distinct closed in-neighborhoods within C. As far as we know, it is the very first time that the spectral graph theory has been applied to the identifying codes. We give a new method to obtain an upper bound on l for digraphs. The results obtained here can also be applied to graphs.Peer ReviewedPostprint (author's final draft

Locating and Identifying Codes in Circulant Networks

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words, S is dominating if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are all nonempty and distinct. A set S \subseteq V(G) is called an identifying code in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We study locating and identifying codes in the circulant networks C_n(1,3). For an integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}

Graphs where every k-subset of vertices is an identifying set

CombinatoricsInternational audienceLet $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subsets identifying any at most $\ell$ vertices is considered as well

Minimal identifying codes in trees and planar graphs with large girth

AbstractLet G be a finite undirected graph with vertex set V(G). If v∈V(G), let N[v] denote the closed neighbourhood of v, i.e. v itself and all its adjacent vertices in G. An identifying code in G is a subset C of V(G) such that the sets N[v]∩C are nonempty and pairwise distinct for each vertex v∈V(G). We consider the problem of finding the minimum size of an identifying code in a given graph, which is known to be NP-hard. We give a linear algorithm that solves it in the class of trees, but show that the problem remains NP-hard in the class of planar graphs with arbitrarily large girth