New lower bounds for identifying codes in infinite grids

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 unique. On a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We present new lower bounds for densities of codes for some small values of $r$ in both the square and hexagonal grids.

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r-Identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500277 ◽

2019 ◽

Vol 11 (02) ◽

pp. 1950027

Author(s):

R. Dhanalakshmi ◽

C. Durairajan

Keyword(s):

Lower Bounds ◽

Upper Bounds ◽

Identifying Codes ◽

Identifying Code ◽

Construction Techniques ◽

Lee Metric ◽

Hamming Space

We study about monotonicity of [Formula: see text]-identifying codes in binary Hamming space, q-ary Lee space and incomplete hypercube. Also, we give the lower bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in [Formula: see text] with respect to the Lee metric. We prove the existence of [Formula: see text]-identifying code in an incomplete hypercube. Also, we give the construction techniques for [Formula: see text]-identifying codes in the incomplete hypercubes in Secs. 4.1 and 4.2. Using these techniques, we give the tables (see Tables 1–6) of upper bounds for [Formula: see text] where [Formula: see text] is the smallest cardinality among all [Formula: see text]-identifying codes in an incomplete hypercube with [Formula: see text] processors. Also, we give the exact values of [Formula: see text] for small values of [Formula: see text] and [Formula: see text] (see Sec. 4.3).

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A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid

The Electronic Journal of Combinatorics ◽

10.37236/202 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 8

Author(s):

Daniel W. Cranston ◽

Gexin Yu

Keyword(s):

Lower Bound ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Hexagonal Grid ◽

Identifying Codes ◽

Identifying Code ◽

Vertex Set

Given a graph $G$, an identifying code ${\cal D}\subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap{\cal D}$ and $N[v_2]\cap{\cal D}$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.413793$.

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