Locating domination in bipartite graphs and their complements

Abstract

A set S of vertices of a graph G is distinguishing if the sets of neighbors in S for every pair of vertices not in S are distinct. A locating-dominating set of G is a dominating distinguishing set. The location-domination number of G, λ(G), is the minimum cardinality of a locating-dominating set. In this work we study relationships between λ(G) and λ(G) for bipartite graphs. The main result is the characterization of all connected bipartite graphs G satisfying λ(G) = λ(G) + 1. To this aim, we define an edge-labeled graph G S associated with a distinguishing set S that turns out to be very helpful.

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

“…The minimum cardinality of such a set, denoted by γ L (G), is the location-domination number of G. There is also an extensive literature on γ L (G) studying multiple aspects: complexity [13,17], specific families [14,16,25,29,31], bounds [2,15,22,40], and approximation algorithms [45]. Clearly, an LD-set is an MLD-set, and so it is also a resolving set; consequently,…”

Improved Lower Bound for Locating-Dominating Codes in Binary Hamming Spaces

“…Lemma 4 has already shown that when n ≥ 13, only the special codewords have share greater than n/2 + 1 + 1/(n 2 − 5n) − 2/(3n). We will proceed by first showing that if we have multiple F n−1 -fathers in B 2 (0), then, after applying Rule 1, we have s(c) < n/2 + 1 − 1/(n − 1), which is less than n/2 + 1 + 1/(n 2 − 5n) − 2/(3n) for each c ∈ I(0) by (5). After that we use similar deduction to show that |I 3 (0)| = n. Finally, we implement some rules to shift share into v and out of v. Therefore, e 3 + e 4 + e n is special.…”

“…Now, since the sensor placement is done using the locating-dominating code, we can deduce the location of the object just by considering which sensors are sending the alarm. The topic of locating-dominating codes has attracted a lot of attention recently, see [3,5,7,10,11]. For more papers in the field consult, the bibliography [13].…”

On Locating-Dominating Set of Regular Graphs

“…Charon et al [3] have proved that determining locatingdominating number of a graph is NP-complete problem which is reduced from 3-SAT. However, some results for certain classes of graphs have been obtained, such as paths [2], cycles [4], stars [5], complete graphs [6], bipartite graphs [7,8], complete multipartite graphs [5], wheels [7], twin-free graphs [9,10], and hypergraphs [11]. In [12], Balbuena et al investigated a locating-dominating set of graphs with girth at least 5.…”