Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1010.2985

Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets

Florent Foucaud·Reza Naserasr·Aline Parreau
View original source ↗Read source document ↗Export metadata ↓

Abstract

An identifying code of a (di)graph $G$ is a dominating subset $C$ of the vertices of $G$ such that all distinct vertices of $G$ have distinct (in)neighbourhoods within $C$. In this paper, we classify all finite digraphs which only admit their whole vertex set in any identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well known theorem of A. Bondy on set systems we classify the extremal cases for this theorem.

Explore related subjects

cs.DMmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Florent Foucaud, Reza Naserasr, Aline Parreau. 2012-02-15. Characterizing extremal digraphs for identifying codes and extremal cases of Bondy's theorem on induced subsets. https://doi.org/10.1007/s00373-012-1136-4

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Factorisability of Low Dimensional Non-Negative Integer Matrices

We consider the problem of determining if a given two-dimensional nonnegative integer matrix $M$ is the product of two such matrices, excluding trivial units. A matrix $M$ with no such factorisation is called prime and therefore belongs to the minimal (infinite rank) generator of $2 \times 2$ matrices over the natural numbers, otherwise it is called composite. We also consider the problem of finding a (non-unique) factorisation of a composite matrix. Our results have applications in computational group theory and the theory of codes, where such matrices are called incidence matrices. We analyse the complexity of primality and finding a factorisation for a composite matrix, providing a first efficient algorithm.

cs.DM↗
arXiv2026

Three Hardness Results for Graph Similarity Problems

Notions of graph similarity provide alternative perspective on the graph isomorphism problem and vice-versa. In this paper, we consider measures of similarity arising from mismatch norms as studied in Gervens and Grohe: the edit distance $δ_{\mathcal{E}}$, and the metrics arising from $\ell_p$-operator norms, which we denote by $δ_p$ and $δ_{|p|}$. We address the following question: can these measures of similarity be used to design polynomial-time approximation algorithms for graph isomorphism? We show that computing an optimal value of $δ_{\mathcal{E}}$ is \NP-hard on pairs of graphs with the same number of edges. In addition, we show that computing optimal values of $δ_p$ and $δ_{|p|}$ is \NP-hard even on pairs of $1$-planar graphs with the same degree sequence and bounded degree. These two results improve on previous known ones, which did not examine the restricted case where the pairs of graphs are required to have the same number of edges. Finally, we study similarity problems on strongly regular graphs and prove some near optimal inequalities with interesting consequences on the computational complexity of graph and group isomorphism.

cs.DM↗