The identifying code, the locating-dominating, the open locating-dominating and the locating total-dominating problems under some graph operations

International audienceThe problems of determining minimum identifying, locating-dominating, open locating-dominating or locating total-dominating codes in a graph G are variations of the classical minimum dominating set problem in G and are all known to be hard for general graphs. A typical line of attack is therefore to determine the cardinality of minimum such codes in special graphs. In this work we study the change of minimum such codes under three operations in graphs: adding a universal vertex, taking the generalized corona of a graph, and taking the square of a graph. We apply these operations to paths and cycles which allows us to provide minimum codes in most of the resulting graph classes

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

On location, domination and information retrieval

The thesis is divided into two main branches: identifying and locatingdominating codes, and information retrieval. The former topics are motivated by the aim to locate objects in sensor networks (or other similar applications) and the latter one by the need to retrieve information in memories such as DNA data storage systems. Albeit the underlying applications, the study on these topics mainly belongs to discrete mathematics; more specically, to the elds of coding and graph theory. The sensor networks are usually represented by graphs where vertices represent the monitored locations and edges the connections between the locations. Moreover, the locations of the sensors are determined by a code. Furthermore, the desired properties of the sensor network are deeply linked with the properties of the underlying code. The number of errors in reading the data is abundant in the DNA data storage systems. In particular, there can occur more errors than a reasonable error-correcting code can handle. However, this problem is somewhat oset by the possibility to obtain multiple approximations of the same information from the data storage. Hence, the information retrieval process can be modelled by the Levenshtein's channel model, where a message is sent through multiple noisy channels and multiple outputs are received. In the rst two papers of the thesis, we introduce and study the new concepts of self- and solid-locating-dominating codes as a natural analogy to self-identifying codes with respect to locating-dominating codes. The rst paper introduces these new codes and considers them in some graphs such as the Hamming graphs. Then, in the second paper, we broaden our view on the topic by considering graph theoretical questions. We give optimal codes in multiple dierent graph classes and some more general results using concepts such as the Dilworth number and graph complements. The third paper focuses on the q-ary Hamming spaces. In particular, we disprove a conjecture proposed by Goddard and Wash related to identifying codes. In the fourth paper, we return to self- and solid-locating-dominating codes and give optimal codes in some graph classes and consider their densities in innite graphs. In the fth paper, we consider information retrieval in memories; in particular, the Levenshtein's channel model. In the channel model, we transmit some codeword belonging to the binary Hamming space through multiple identical channels. With the help of multiple dierent outputs, we give a list of codewords which may have been sent. In the paper, we study the number of channels required to have a rather small (constant) list size when the properties of the channels, the code and the dimension of the Hamming space are xed. In particular, we give an exact relation between the number of channels and the asymptotic value of the maximum list size.Väitöskirja käsittelee kahta aihetta: identioivia ja paikantavia peittokoodeja sekä tiedon noutamista muistista. Ensimmäisen aiheen motivaationa on objektien paikantaminen sensoriverkoista (sekä muut samankaltaiset sovellukset) ja jälkimmäisen tiedonnouto DNA-muisteista. Näiden aiheiden tutkimus kuuluu diskreettiin matematiikkaan, täsmällisemmin koodaus- ja graa-teoriaan. Sensoriverkkoja kuvataan yleensä graafeilla, joissa solmut esittävät tarkkailtuja kohteita ja viivat yhteyksiä näiden kohteiden välillä. Edelleen sensorien paikat määräytyvät annetun koodin perusteella. Tästä johtuen sensoriverkon halutut ominaisuudet pohjautuvat vahvasti alla olevaan koodiin. Luettaessa tietoa DNA-muisteista tapahtuvien virheiden määrä saattaa olla erittäin suuri; erityisesti suurempi kuin kiinnitetyn virheitä korjaavan koodin korjauskyky. Toisaalta tilanne ei ole aivan näin ongelmallinen, sillä DNA-muisteista voidaan saada useita eri arvioita muistiin tallennetusta tiedosta. Näistä syistä johtuen tietojen noutamista DNA-muisteista voidaan mallintaa käyttäen Levenshteinin kanavamallia. Kanavamallissa yksi viesti lähetetään useiden häiriöisten kanavien kautta ja näin vastaanotetaan useita viestejä (yksi jokaisesta kanavasta). Väitöskirjan kahdessa ensimmäisessä julkaisussa esitellään ja tutkitaan uusia paikantavien peittokoodien luokkia, jotka pohjautuvat aiemmin tutkittuihin itse-identioiviin koodeihin. Ensimmäisessä julkaisussa on esitelty nämä koodiluokat sekä tutkittu niitä joissain graafeissa kuten Hammingin graafeissa. Tämän jälkeen toisessa julkaisussa käsitellään yleisiä graa-teoreettisia kysymyksiä. Julkaisussa esitetään optimaaliset koodit useille graaperheille sekä joitain yleisempiä tuloksia käyttäen mm. Dilworthin lukua sekä graakomplementteja. Kolmas julkaisu keskittyy q-arisiin Hammingin avaruuksiin. Erityisesti julkaisussa todistetaan vääräksi Goddardin ja Washin aiemmin esittämä identioivia koodeja koskeva otaksuma. Neljäs artikkeli käsittelee jo kahdessa ensimmäisessä artikkelissa esiteltyjä paikantavien peittokoodien luokkia. Artikkeli esittää optimaalisia koodeja useille graaperheille sekä käsittelee äärettömiä graafeja. Viides artikkeli käsittelee tiedonnoutoa ja erityisesti Levenshteinin kanavamallia. Kanavamallissa binääriseen Hammingin avaruuteen kuuluva koodisana lähetetään useiden identtisten kanavien läpi. Näistä kanavista vastaanotetaan useita eri arvioita lähetetystä koodisanasta ja rakennetaan lista mahdollisesti lähetetyistä sanoista. Artikkelissa tutkitaan kuinka monta kanavaa tarvitaan, jotta tämän listan koko on pieni (vakio), kun kanavien ominaisuudet, koodi ja Hammingin avaruuden dimensio on kiinnitetty. Erityisesti löydetään täsmällinen suhde kanavien lukumäärän ja asymptoottisesti maksimaalisen listan koon välille

Optimization Approaches for Open-Locating Dominating Sets

An Open Locating-Dominating Set (OLD set) is a subset of vertices in a graph such that every vertex in the graph has a neighbor in the OLD set and every vertex has a unique set of neighbors in the OLD set. This can also represent where sensors, capable of detecting an event occurrence at an adjacent vertex, could be placed such that one could always identify the location of an event by the specific vertices that indicated an event occurred in their neighborhood. By the open neighborhood construct, which differentiates OLD sets from identifying codes, a vertex is not able to report if it is the location of the event. This construct provides a robustness over identifying codes and opens new applications such as disease carrier and dark actor identification in networks. This work explores various aspects of OLD sets, beginning with an Integer Linear Program for quickly identifying the optimal OLD set on a graph. As many graphs do not admit OLD sets, or there may be times when the total size of the set is limited by an external factor, a concept called maximum covering OLD sets is developed and explored. The coverage radius of the sensors is then expanded in a presentation of Mixed-Weight OLD sets where sensors can cover more than just adjacent vertices. Finally, an application is presented to optimally monitor criminal and terrorist networks using OLD sets and related concepts to identify the optimal set of surveillance targets

On Identifying and Locating-Dominating Codes in the Infinite King Grid

Siirretty Doriast

Identifying Codes and Domination in the Product of Graphs

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code in a graph $G$ is denoted \gid(G). We consider identifying codes of the direct product $K_n \times K_m$. In particular, we answer a question of Klav\v{z}ar and show the exact value of \gid(K_n \times K_m). It was recently shown by Gravier, Moncel and Semri that for the Cartesian product of two same-sized cliques \gid(K_n \Box K_n) = \lfloor{\frac{3n}{2}\rfloor}. Letting $m \ge n \ge 2$ be any integers, we show that \IDCODE(K_n \cartprod K_m) = \max\{2m-n, m + \lfloor n/2 \rfloor\}. Furthermore, we improve upon the bounds for \IDCODE(G \cartprod K_m) and explore the specific case when $G$ is the Cartesian product of multiple cliques. Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $\pi$ of $V(G)$, the permutation graph $\pi G$ is constructed by joining $u \in V(G^1)$ to $\pi(u) \in V(G^2)$ for all $u \in V(G^1)$. The graph $G$ is said to be a universal fixer if the domination number of $\pi G$ is equal to the domination number of $G$ for all $\pi$ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. We prove the conjecture

Progress towards the two-thirds conjecture on locating-total dominating sets

We study upper bounds on the size of optimum locating-total dominating sets in graphs. A set $S$ of vertices of a graph $G$ is a locating-total dominating set if every vertex of $G$ has a neighbor in $S$, and if any two vertices outside $S$ have distinct neighborhoods within $S$. The smallest size of such a set is denoted by $\gamma^L_t(G)$. It has been conjectured that $\gamma^L_t(G)\leq\frac{2n}{3}$ holds for every twin-free graph $G$ of order $n$ without isolated vertices. We prove that the conjecture holds for cobipartite graphs, split graphs, block graphs, subcubic graphs and outerplanar graphs