A locating-total dominating set of a graph G = (V (G), E(G)) with no isolated vertex is a set S ⊆ V (G) such that every vertex of V (G) is adjacent to a vertex of S and for every pair of distinct vertices u and v in V (G) − S, N(u) ∩ S = N(v) ∩ S. Let γ t (G) be the minimum cardinality of a locating-total dominating set of G. A graph G is said to be locating-total domination vertex critical if ...

In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.

A set D of vertices in a graph G = (V, E) is a locating-dominating set (LDS) if for every two vertices u, v of V − D the sets N(u) ∩ D and N(v) ∩ D are non-empty and different. The locating-domination number γL(G) is the minimum cardinality of a LDS of G, and the upper locating-domination number, ΓL(G) is the maximum cardinality of a minimal LDS of G. We present different bounds on ΓL(G) and γL...

We model a problem about networks built from wireless devices using identifying and locating-dominating codes in unit disk graphs. It is known that minimising the size of an identifying code is NP-complete even for bipartite graphs. First, we improve this result by showing that the problem remains NP-complete for bipartite planar unit disk graphs. Then, we address the question of the existence ...

Let c be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) be an ordered partition of V (G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple cΠ(v) := (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) | x ∈ Vi}, 1 ≤ i ≤ k. If distinct vertices have distinct color codes, then ...

‎in this paper we introduce mixed unitary cayley graph $m_{n}$‎ ‎$(n>1)$ and compute its eigenvalues‎. ‎we also compute the energy of‎ ‎$m_{n}$ for some $n$‎.