9 research outputs found
Automated Discharging Arguments for Density Problems in Grids
Discharging arguments demonstrate a connection between local structure and
global averages. This makes it an effective tool for proving lower bounds on
the density of special sets in infinite grids. However, the minimum density of
an identifying code in the hexagonal grid remains open, with an upper bound of
and a lower bound of . We present a new, experimental framework for producing discharging
arguments using an algorithm. This algorithm replaces the lengthy case analysis
of human-written discharging arguments with a linear program that produces the
best possible lower bound using the specified set of discharging rules. We use
this framework to present a lower bound of on
the density of an identifying code in the hexagonal grid, and also find several
sharp lower bounds for variations on identifying codes in the hexagonal,
square, and triangular grids.Comment: This is an extended abstract, with 10 pages, 2 appendices, 5 tables,
and 2 figure
Open-independent, Open-locating-dominating Sets
A distinguishing set for a graph G = (V, E) is a dominating set D, each vertex being the location of some form of a locating device, from which one can detect and precisely identify any given "intruder" vertex in V(G). As with many applications of dominating sets, the set might be required to have a certain property for <D>, the subgraph induced by D (such as independence, paired, or connected). Recently the study of independent locating-dominating sets and independent identifying codes was initiated. Here we introduce the property of open-independence for open-locating-dominating sets
Locating and Identifying Codes in Circulant Networks
A set S of vertices of a graph G is a dominating set of G if every vertex u
of G is either in S or it has a neighbour in S. In other words, S is dominating
if the sets S\cap N[u] where u \in V(G) and N[u] denotes the closed
neighbourhood of u in G, are all nonempty. A set S \subseteq V(G) is called a
locating code in G, if the sets S \cap N[u] where u \in V(G) \setminus S are
all nonempty and distinct. A set S \subseteq V(G) is called an identifying code
in G, if the sets S\cap N[u] where u\in V(G) are all nonempty and distinct. We
study locating and identifying codes in the circulant networks C_n(1,3). For an
integer n>6, the graph C_n(1,3) has vertex set Z_n and edges xy where x,y \in
Z_n and |x-y| \in {1,3}. We prove that a smallest locating code in C_n(1,3) has
size \lceil n/3 \rceil + c, where c \in {0,1}, and a smallest identifying code
in C_n(1,3) has size \lceil 4n/11 \rceil + c', where c' \in {0,1}
An optimal strongly identifying code in the infinite triangular grid
Assume that G = (V, E) is an undirected graph, and C subset of V. For every v is an element of V, we denote by I(v) the set of all elements of C that are within distance one from v. If the sets I(v){v} for v is an element of V are all nonempty, and, moreover, the sets {I(v), I(v){v}} for v is an element of V are disjoint, then C is called a strongly identifying code. The smallest possible density of a strongly identifying code in the infinite triangular grid is shown to be 6/19
Optimal local identifying and local locating-dominating codes
We introduce two new classes of covering codes in graphs for every positive
integer . These new codes are called local -identifying and local
-locating-dominating codes and they are derived from -identifying and
-locating-dominating codes, respectively. We study the sizes of optimal
local 1-identifying codes in binary hypercubes. We obtain lower and upper
bounds that are asymptotically tight. Together the bounds show that the cost of
changing covering codes into local 1-identifying codes is negligible. For some
small optimal constructions are obtained. Moreover, the upper bound is
obtained by a linear code construction. Also, we study the densities of optimal
local 1-identifying codes and local 1-locating-dominating codes in the infinite
square grid, the hexagonal grid, the triangular grid, and the king grid. We
prove that seven out of eight of our constructions have optimal densities
Optimal bounds on codes for location in circulant graphs
Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1,2,3,...,r) over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs Cn(1,d) for d=3 and proposed as an open question the case of d>3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1,d), Cn(1,d-1,d) and Cn(1,d-1,d,d+1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1,3) and Cn(1,4)
Lokaali identifiointi graafeissa
Tässä tutkielmassa esitellään kaksi uutta peittokoodien luokkaa - lokaalisti identifioivat koodit ja lokaalisti paikallistavat-dominoivat koodit - ja todistetaan näihin liityviä tuloksia eri graafeissa. Tuloksia verrataan vastaaviin tunnettuihin tuloksiin koskien identifioivia ja paikallistavia-dominoivia koodeja. Myös vertailua peittokoodeihin tehdään.
Tutkielma alkaa lyhyellä johdannolla aiheeseen, jonka jälkeen esitellään suurin osa tarvittavista käsitteistä ja määritelmistä toisessa luvussa. Kolmannessa luvussa tutkitaan lyhyesti lokaalisti identifioivia koodeja poluissa ja sykleissä sekä erityisesti niiden suhdetta identifioiviin koodeihin samaisissa graafeissa. Luvussa neljä tarkastellaan identifiointia binäärisissä hyperkuutioissa ja todistetaan tuloksia lokaalisti 1-identifioiville koodeille näissä graafeissa. Viimeisessä luvussa siirrytään joihinkin äärrettömiin hiloihin, joissa tutkitaan lokaalisti 1-identifioivia ja lokaalisti 1-paikallistavia-dominoivia koodeja