Petru Valicov

Extremal graphs for the identifying code problem

An identifying code of a graph G is a dominating set C such that every vertex x of G is distinguished from other vertices by the set of vertices in C that are at distance at most 1 from x. The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A. Lobstein that if a graph on n vertices with at least one edge admits an identifying code, then a minimal identifying code has size at most n-1. They introduced classes of graphs whose smallest identifying code is of size n-1. Few conjectures were formulated to classify the class of all graphs whose minimum identifying code is of size n-1. In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided.

IDENTIFYING CODES IN LINE GRAPHS

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

On strong edge-colouring of subcubic graphs

\ID(G)

Strong edge-colouring and induced matchings

denotes the size of a minimum identifying code of an identifiable graph

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

G

Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

, we show that the usual bound

MPQ-trees for the Orthogonal Packing Problem

\ID(G)\ge \lceil\log_2(n+1)\rceil

Planar Digraphs without Large Acyclic Sets

, where

Strong edge-coloring of ( 3 , Δ ) -bipartite graphs

n

Edge identifying codes

denotes the order of