Extremal cardinalities for identifying and locating-dominating codes in graphs

Extremal cardinalities for identifying and locating-dominating codes in graphs

Consider 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 centred at v, i.e., the set of all vertices linked to v by a path of at most r edges. If for all vertices v ∈ V (respectively, v ∈ V \C), the setsBr(v)∩C are all nonempty and different, then we call C an r-identifying code (respectively,...

Locating dominating codes: Bounds and extremal cardinalities

In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our more significant contributions on λ-codes and η-codes concerning concerning bounds, extremal val...

Possible cardinalities for locating-dominating codes in graphs

Possible cardinalities for identifying codes in graphs

Hardness results and approximation algorithms for identifying codes and locating-dominating codes in graphs

In a graph G = (V, E), an identifying code of G (resp. a locating-dominating code of G) is a subset of vertices C ⊆ V such that N [v]∩C 6= ∅ for all v ∈ V , and N [u] ∩C 6= N [v]∩C for all u 6= v, u, v ∈ V (resp. u, v ∈ V r C), where N [u] denotes the closed neighbourhood of v, that is N [u] = N(u) ∪ {u}. These codes model fault-detection problems in multiprocessor systems and are also used for...