Reza Naserasr

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

Homomorphisms of planar signed graphs to signed projective cubes

\ID(G)

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

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

On a new reformulation of Hadwiger's conjecture

G

Mapping Planar Graphs into Projective Cubes

, we show that the usual bound

The complexity of signed graph and edge-coloured graph homomorphisms

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

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

, where

The Complexity of Homomorphisms of Signed Graphs and Signed Constraint Satisfaction

n

Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy's Theorem on Induced Subsets

denotes the order of