Extremal cardinalities for identifying and locating-dominating codes in graphs
نویسندگان
چکیده
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, an r-locating-dominating code). We study the extremal values of the cardinality of a minimum r-identifying or r-locating-dominating code in any connected undirected graphG having a given number, n, of vertices. It is known that aminimum r-identifying code contains at least log2(n+1) vertices; we establish in particular that such a code contains at most n− 1 vertices, and we prove that these two bounds are reached. The same type of results are given for locating-dominating codes. © 2006 Elsevier B.V. All rights reserved.
منابع مشابه
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...
متن کاملLocating-Dominating Sets and Identifying Codes in Graphs of Girth at least 5
Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and o...
متن کاملIdentifying and Locating-Dominating Codes in (Random) Geometric Networks
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 ...
متن کاملHardness 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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Discrete Mathematics
دوره 307 شماره
صفحات -
تاریخ انتشار 2007