Related papers: Progress on Self Identifying Codes
In this paper, we study the redundancy of linear codes with graph constraints. First we consider linear parity check codes based on bipartite graphs with diversity and with generalized graph constraints. We describe sufficient conditions on…
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised…
Given a graph $G$, an identifying code $C \subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap C$ and $N[v_2]\cap C$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and…
Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In particular, there are a lot of results regarding the binary hypercubes, that is, the Hamming graphs $K_2^n$. In…
The distinguishing number of a graph was introduced by Albertson and Collins as a measure of the amount of symmetry contained in the graph. Tymoczko extended this definition to faithful group actions on sets; taking the set to be the vertex…
Scalable quantum computing and communication requires the protection of quantum information from the detrimental effects of decoherence and noise. Previous work tackling this problem has relied on the original circuit model for quantum…
In this paper, we investigate the problem of covering the vertices of a graph associated to a finite vector space as introduced by Das \cite{Das}, such that we can uniquely identify any vertex by examining the vertices that cover it. We use…
An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the…
In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpinski graphs. We compute the minimum size of such codes in Sierpinski graphs.
The singularity degree of a semidefinite programming problem is the smallest number of facial reduction steps to make the problem strictly feasible. We introduce two new graph parameters, called the singularity degree and the nondegenerate…
An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and different. Here, we provide an overview on codes for the…
In 1985, Chv\'{a}tal introduced the concept of star cutsets as a means to investigate the properties of perfect graphs, which inspired many researchers to study cutsets with some specific structures, for example, star cutsets, clique…
Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal…
In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique $O(\log n)$-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms)…
The concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times. The local crossing number of a graph is the…
The independent domination number $i(G)$ of a graph $G$ is the minimum cardinality of a maximal independent set of $G$, also called an $i(G)$-set. The $i$-graph of $G$, denoted $\mathcal{I}(G)$, is the graph whose vertices correspond to the…
The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error…
Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the…
The graph isomorphism problem is of practical importance, as well as being a theoretical curiosity in computational complexity theory in that it is not known whether it is $NP$-complete or $P$. However, for many graphs, the problem is…
In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve…