A scheme to construct distance-three codes using latin squares, with applications to the n-cube

This note presents a technical improvement to an upper bound in “The domination number of exchanged hypercubes” [Inf. Process. Lett. 114 (4) (2014) 159–162] by Klavžar and Ma.

If r⩾1, and m and n are each a multiple of (r+1)²+r², then each isomorphic component of C_m×C_n admits of a vertex partition into (r+1)²+r² perfect r-dominating sets. The result induces a dense packing of C_m×C_n by means of vertex-disjoint subgraphs, each isomorphic to a diagonal array. Areas of applications include efficient resource placement in a diagonal mesh and error-correcting codes.

Let $\text{k⩾2,}\text{n=2}{}^{\mathrm{k}}\text{+1,}$ and let m₀,…,m_k−1 each be a multiple of n. The graph C_m0×⋯×C_mk−1 consists of isomorphic connected components, each of which is (n−1)-regular and admits of a vertex partition into n smallest independent dominating sets. Accordingly, (independent) domination number of each connected component of this graph is equal to (1/n)th of the number of vertices in it.

A set of vertices S is subverted from a graph G by removing the closed neighborhood N[S] from G. We denote the survival subgraph of the vertex subversion strategy S by G/S. The vertex-neighbor-integrity of G is defined to be VNI(G)=min_S⊆V(G){|S|+ω(G/S)}, where ω(H) is the order of the largest connected component in the graph H. The graph parameter VNI was introduced by Cozzens and Wu [3] to measure the vulnerability of a spy network. Cozzens and Wu showed that the VNI of paths, cycles, trees and powers of paths on n vertices are all on the order of $\text{n}$. Here we prove that the VNI of any member of a family of magnifier graphs is linear in the order of the graph. We also find upper and lower bounds on the VNI of hypercubes. Finally, we show that the decision problem corresponding to computing the vertex-neighbor-integrity of a graph is NP-complete.