On two variations of identifying codes

Identifying codes have been introduced in 1998 to model fault-detection in multiprocessor systems. In this paper, we introduce two variations of identifying codes: weak codes and light codes. They correspond to fault-detection by successive rounds. We give exact bounds for those two definitions for the family of cycles

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $\zeta (G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable

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

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

On three domination numbers in block graphs

The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view. Hence, a typical line of attack for these problems is to determine lower and upper bounds for minimum codes in special graphs. In this work we study the problem of determining the cardinality of minimum codes in block graphs (that are diamond-free chordal graphs). We present for all three codes lower and upper bounds as well as block graphs where these bounds are attained

Identifying codes of corona product graphs

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$

Mixed-Weight Open Locating-Dominating Sets

The detection and location of issues in a network is a common problem encompassing a wide variety of research areas. Location-detection problems have been studied for wireless sensor networks and environmental monitoring, microprocessor fault detection, public utility contamination, and finding intruders in buildings. Modeling these systems as a graph, we want to find the smallest subset of nodes that, when sensors are placed at those locations, can detect and locate any anomalies that arise. One type of set that solves this problem is the open locating-dominating set (OLD-set), a set of nodes that forms a unique and nonempty neighborhood with every node in the graph. For this work, we begin with a study of OLD-sets in circulant graphs. Circulant graphs are a group of regular cyclic graphs that are often used in massively parallel systems. We prove the optimal OLD-set size for two circulant graphs using two proof techniques: the discharging method and Hall\u27s Theorem. Next we introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the OLD-set. The mixed-weight OLD-set allows nodes in the graph to have different weights, representing systems that use sensors of varying strengths. This is a novel approach to the study of location-detection problems. We show that the decision problem for the minimum mixed-weight OLD-set, for any weights up to positive integer d, is NP-complete. We find the size of mixed-weight OLD-sets in paths and cycles for weights 1 and 2. We consider mixed-weight OLD-sets in random graphs by providing probabilistic bounds on the size of the mixed-weight OLD-set and use simulation to reinforce the theoretical results. Finally, we build and study an integer linear program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our results for mixed-weight OLD-sets in random graphs to random geometric graphs by estimating the probabilistic upper bound for the size of the set