9 research outputs found
Improved Bounds for -Identifying Codes of the Hex Grid
For any positive integer , an -identifying code on a graph is a set
such that for every vertex in , the intersection of the
radius- closed neighborhood with is nonempty and pairwise distinct. For
a finite graph, the density of a code is , which naturally extends
to a definition of density in certain infinite graphs which are locally finite.
We find a code of density less than , which is sparser than the prior
best construction which has density approximately .Comment: 12p
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
Identifying codes and locating–dominating sets on paths and cycles
AbstractLet G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define Nr[x]={y∈V:d(x,y)≤r} and Dr(x)=Nr[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V∖D, respectively), Dr(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72–82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987]
Identifying and locating-dominating codes on chains and cycles
AbstractConsider a connected undirected graph G=(V,E), a subset of vertices C⊆V, and an integer r≥1; for any vertex v∈V, let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v∈V (respectively, v∈V ⧹C), the sets Br(v)∩C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles
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
On Vertex Identifying Codes For Infinite Lattices
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes
in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex
Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with
some other resultsComment: 91p