General Bounds for Identifying Codes in Some Infinite Regular Graphs

Abstract

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

Optimal Lower Bound for 2-Identifying Codes in the Hexagonal Grid

“…Assume that share is shifted from (4, 1) according to the rule 5 (see Figure 3(e)). Now we have C ∩ S 1 = {(2, 2), (4, 1)} and (4, 3) ∈ C. Therefore, since at most 1/12 units of share is shifted from (4, 1) according to the rule 2.1, we obtain that no more than 1/6 + 1/12 = 1/4 units of share can be shifted from S 1 to c. Similarly, if share is shifted from (5,1) according to the rule 6, then it can be shown that c receives at most 1/4 units of share from S 1 . In conclusion, at most 1/4 units of share is shifted from the vertices of S 1 to c. Analogously, this statement also holds for the vertices of S 2 .…”

“…Observe then that if u ∈ S 1 shifts share to c according to the rules 1-6 or their modifications, then we have I 2 (u) = {u}. Therefore, if two vertices of S 1 shift share, then one of these vertices is (4, 1) or (5,1). Assume that share is shifted from (4, 1) according to the rule 5 (see Figure 3(e)).…”

“…Therefore, by Lemma 2.1, it is straightforward (albeit tedious) to verify that in all possible casess 2 (c) ≤ s 2 (c) ≤ 19/4. Indeed, we can choose in Lemma 2.1 the set D to consist of c, (2, 0) and a codeword in the symmetric difference B 2 (c) △ B 2 (1,0).…”

“…Later these results have been improved by showing that there exists a 1-identifying code with density 3/7 (see Cohen et al [3]) and that there do not exist 1-identifying codes in G H with density smaller than 12/29 (see Cranston and Yu [4]). For general r ≥ 2, Charon et al [1] showed that each r-identifying code C in G H has D(C) ≥ 2/(5r + 3) if r is even and D(C) ≥ 2/(5r + 2) if r is odd. They also presented a construction for each r ≥ 2 giving an r-identifying code C ⊆ V H with D(C) ∼ 8/(9r).…”

Identifying codes of degree 4 Cayley graphs over Abelian groups

“…Given a vertex v ∈ V the set of codewords which r-cover v are denoted as: Although the aim of this paper is to construct identifying codes over G(M ) graphs, the problem over the infinite mesh will be considered before. Previous papers dealing with the problem of identification over the infinite mesh are, for example, [7], [10] and [6]. Therefore, let us define the infinite mesh as the Cayley graph Cay(Z 2 , E).…”